:: The Measurability of Complex-Valued Functional Sequences
:: by Keiko Narita , Noboru Endou and Yasunari Shidama
::
:: Received December 16, 2008
:: Copyright (c) 2008-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, XBOOLE_0, PROB_1, MEASURE1, PARTFUN1, REAL_1, SUBSET_1,
SEQFUNC, MEASURE6, RELAT_1, FUNCT_1, PBOOLE, TARSKI, SEQ_1, ORDINAL2,
RINFSUP1, MESFUNC8, CARD_1, NAT_1, MESFUNC1, ARYTM_3, SEQ_2, XXREAL_0,
XXREAL_2, SETFAM_1, COMSEQ_1, COMPLEX1, VALUED_1, SUPINF_2, POWER,
ARYTM_1, MESFUNC5, INTEGRA5, MESFUNC2, FINSEQ_1, MESFUNC3, INT_1,
ZFMISC_1, XCMPLX_0, FUNCT_7;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XXREAL_3, XCMPLX_0,
COMPLEX1, XXREAL_0, XREAL_0, XXREAL_2, REAL_1, NAT_1, NAT_D, PROB_1,
RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FINSEQ_1, RFUNCT_3, VALUED_1,
FUNCT_2, SETFAM_1, SUPINF_1, SUPINF_2, SEQ_1, SEQ_2, SEQFUNC, COMSEQ_1,
COMSEQ_2, RINFSUP1, RINFSUP2, MEASURE1, MEASURE6, EXTREAL1, MESFUNC1,
MESFUNC2, MESFUNC5, MESFUNC6, COMSEQ_3, MESFUN6C, MESFUNC8, SEQ_4,
COMPLSP2;
constructors REAL_1, SQUARE_1, MEASURE6, EXTREAL1, MESFUNC2, MESFUNC3,
MESFUNC5, MESFUNC6, MESFUN6C, BINOP_2, RINFSUP1, MESFUNC8, COMSEQ_2,
COMSEQ_3, SUPINF_1, RINFSUP2, SEQFUNC, MESFUNC1, COMPLSP2, MATRIX_5,
NAT_D, RELSET_1, SEQ_4;
registrations NAT_1, MESFUNC8, SUBSET_1, ORDINAL1, NUMBERS, XXREAL_0, XREAL_0,
MEMBERED, VALUED_0, XCMPLX_0, PARTFUN1, FUNCT_2, RELAT_1, SEQ_2,
COMSEQ_3, RELSET_1, XXREAL_3;
requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
definitions TARSKI, XBOOLE_0;
equalities MESFUNC5, COMPLEX1, RINFSUP2, XBOOLE_0, FINSEQ_1, COMPLSP2;
expansions MESFUNC5, COMPLEX1, TARSKI, XBOOLE_0;
theorems MEASURE1, TARSKI, PARTFUN1, FUNCT_1, MESFUNC1, NAT_1, XBOOLE_0,
XBOOLE_1, XCMPLX_0, XREAL_1, MESFUNC5, XXREAL_0, VALUED_1, MESFUNC6,
COMPLEX1, RELAT_1, SQUARE_1, FINSEQ_1, MESFUN6C, ORDINAL1, FUNCT_2,
SEQFUNC, SETFAM_1, RINFSUP1, MESFUNC7, MESFUNC8, COMSEQ_3, MESFUNC3,
NUMBERS, RINFSUP2, XXREAL_2, COMPLSP2, NAT_D, FINSEQ_2, NAT_2, FINSEQ_3,
RELSET_1, XXREAL_3, XREAL_0;
schemes FUNCT_2, PARTFUN2, FINSEQ_1;
begin :: Real-valued Functional Sequences
reserve X for non empty set,
Y for set,
S for SigmaField of X,
M for sigma_Measure of S,
f,g for PartFunc of X,COMPLEX,
r for Real,
k for Real,
n for Nat,
E for Element of S;
definition
let X be non empty set;
let f be Functional_Sequence of X,REAL;
func R_EAL f -> Functional_Sequence of X,ExtREAL equals
f;
coherence
proof
dom f = NAT & for n being Nat holds f.n is PartFunc of X,
ExtREAL by NUMBERS:31,RELSET_1:7,SEQFUNC:1;
hence thesis by SEQFUNC:1;
end;
end;
theorem Th1:
for X be non empty set, f be Functional_Sequence of X,REAL, x be
Element of X holds f#x = (R_EAL f)#x
proof
let X be non empty set;
let f be Functional_Sequence of X,REAL;
let x be Element of X;
now
let r be object;
assume r in rng((R_EAL f)#x);
then consider n be object such that
A1: n in NAT and
A2: ((R_EAL f)#x).n = r by FUNCT_2:11;
reconsider n as Element of NAT by A1;
r = ((R_EAL f).n).x by A2,MESFUNC5:def 13
.= (R_EAL(f.n)).x
.= (f.n).x;
hence r in REAL by XREAL_0:def 1;
end;
then rng((R_EAL f)#x) c= REAL;
then reconsider RFx = (R_EAL f)#x as sequence of REAL by FUNCT_2:6;
reconsider RFx as Real_Sequence;
now
let n be object;
assume n in NAT;
then reconsider n1 = n as Element of NAT;
RFx.n = ((R_EAL f).n1).x by MESFUNC5:def 13
.= (R_EAL(f.n1)).x;
hence RFx.n = (f#x).n by SEQFUNC:def 10;
end;
hence thesis by FUNCT_2:12;
end;
registration
let X be non empty set, f be Function of X,REAL;
cluster R_EAL f -> total;
coherence;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func inf f -> PartFunc of X,ExtREAL equals
inf R_EAL f;
coherence;
end;
theorem Th2:
for X being non empty set, f being Functional_Sequence of X,REAL
holds for x be Element of X st x in dom inf f holds (inf f).x = inf rng R_EAL(f
#x)
proof
let X be non empty set, f be Functional_Sequence of X,REAL;
let x be Element of X;
assume x in dom inf f;
then (inf f).x = inf((R_EAL f)#x) by MESFUNC8:def 3;
hence thesis by Th1;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func sup f -> PartFunc of X,ExtREAL equals
sup R_EAL f;
coherence;
end;
theorem Th3:
for X being non empty set, f being Functional_Sequence of X,REAL
holds for x be Element of X st x in dom sup f holds (sup f).x = sup rng R_EAL(f
#x)
proof
let X be non empty set, f be Functional_Sequence of X,REAL;
let x be Element of X;
assume x in dom sup f;
then (sup f).x = sup((R_EAL f)#x) by MESFUNC8:def 4;
hence thesis by Th1;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func inferior_realsequence f -> with_the_same_dom Functional_Sequence of X,
ExtREAL equals
inferior_realsequence R_EAL f;
coherence;
end;
theorem Th4:
for X be non empty set, f being Functional_Sequence of X,REAL, n
being Nat holds dom((inferior_realsequence f).n) = dom(f.0) & for x
be Element of X st x in dom((inferior_realsequence f).n) holds ((
inferior_realsequence f).n).x=(inferior_realsequence R_EAL(f#x)).n
proof
let X be non empty set;
let f be Functional_Sequence of X,REAL;
let n be Nat;
set IF = inferior_realsequence f;
dom(IF.n) = dom((R_EAL f).0) by MESFUNC8:def 5
.= dom R_EAL(f.0);
hence dom((inferior_realsequence f).n) = dom(f.0);
hereby
let x be Element of X;
assume x in dom(IF.n);
then
A1: (IF.n).x = (inferior_realsequence((R_EAL f)#x)).n by MESFUNC8:def 5
.= inf( ((R_EAL f)#x)^\n ) by RINFSUP2:27;
(R_EAL f)#x = f#x by Th1;
hence (IF.n).x = (inferior_realsequence R_EAL(f#x)).n by A1,RINFSUP2:27;
end;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func superior_realsequence f -> with_the_same_dom Functional_Sequence of X,
ExtREAL equals
superior_realsequence R_EAL f;
coherence;
end;
theorem Th5:
for X be non empty set, f being Functional_Sequence of X,REAL, n
being Nat holds dom((superior_realsequence f).n) = dom(f.0) & for x
be Element of X st x in dom((superior_realsequence f).n) holds ((
superior_realsequence f).n).x=(superior_realsequence R_EAL(f#x)).n
proof
let X be non empty set;
let f be Functional_Sequence of X,REAL;
let n be Nat;
set SF = superior_realsequence f;
thus dom((superior_realsequence f).n) = dom(f.0) by MESFUNC8:def 6;
hereby
let x be Element of X;
assume x in dom(SF.n);
then (SF.n).x = (superior_realsequence((R_EAL f)#x)).n by MESFUNC8:def 6;
hence (SF.n).x = (superior_realsequence R_EAL(f#x)).n by Th1;
end;
end;
theorem
for f be Functional_Sequence of X,REAL, x be Element of X st x in dom(
f.0) holds (inferior_realsequence f)#x = inferior_realsequence R_EAL(f#x)
proof
let f be Functional_Sequence of X,REAL;
let x be Element of X;
set F = inferior_realsequence f;
assume
A1: x in dom (f.0);
now
let n be Element of NAT;
dom(F.n) = dom (f.0) & (F#x).n = (F.n).x by Th4,MESFUNC5:def 13;
hence (F#x).n = (inferior_realsequence R_EAL(f#x)).n by A1,Th4;
end;
hence thesis by FUNCT_2:63;
end;
registration
let X be non empty set, f be with_the_same_dom Functional_Sequence of X,REAL;
cluster R_EAL f -> with_the_same_dom;
coherence
proof
for n,m be Nat holds dom((R_EAL f).n) = dom((R_EAL f).m) by MESFUNC8:def 2;
hence thesis by MESFUNC8:def 2;
end;
end;
theorem Th7:
for X be non empty set, f be with_the_same_dom
Functional_Sequence of X,REAL for S be SigmaField of X, E be Element of S, n be
Nat st f.n is_measurable_on E holds (R_EAL f).n is_measurable_on E
proof
let X be non empty set, f be with_the_same_dom Functional_Sequence of X,REAL;
let S be SigmaField of X, E be Element of S, n be Nat;
assume f.n is_measurable_on E;
then R_EAL(f.n) is_measurable_on E by MESFUNC6:def 1;
hence thesis;
end;
theorem
for X be non empty set, f being Functional_Sequence of X,REAL, n
being Nat holds (R_EAL f)^\n = R_EAL(f^\n);
theorem
for f be with_the_same_dom Functional_Sequence of X,REAL, n be Nat
holds (inferior_realsequence f).n = inf(f^\n)
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, n be Nat;
n in NAT by ORDINAL1:def 12;
then (inferior_realsequence R_EAL f).n = inf((R_EAL f)^\n) by MESFUNC8:8;
hence thesis;
end;
theorem
for f be with_the_same_dom Functional_Sequence of X,REAL, n be Nat
holds (superior_realsequence f).n = sup(f^\n)
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, n be Nat;
n in NAT by ORDINAL1:def 12;
then (superior_realsequence R_EAL f).n = sup((R_EAL f)^\n) by MESFUNC8:9;
hence thesis;
end;
theorem Th11:
for f be Functional_Sequence of X,REAL, x be Element of X st x
in dom(f.0) holds (superior_realsequence f)#x = superior_realsequence R_EAL(f#x
)
proof
let f be Functional_Sequence of X,REAL, x be Element of X;
set F = superior_realsequence f;
assume
A1: x in dom(f.0);
now
let n be Element of NAT;
dom(F.n) = dom(f.0) & (F#x).n = (F.n).x by Th5,MESFUNC5:def 13;
hence (F#x).n =(superior_realsequence R_EAL(f#x)).n by A1,Th5;
end;
hence thesis by FUNCT_2:63;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func lim_inf f -> PartFunc of X,ExtREAL equals
lim_inf R_EAL f;
coherence;
end;
theorem Th12:
for X be non empty set, f be Functional_Sequence of X,REAL holds
for x be Element of X st x in dom lim_inf f holds (lim_inf f).x = lim_inf R_EAL
(f#x)
proof
let X be non empty set, f be Functional_Sequence of X,REAL;
let x be Element of X;
assume x in dom lim_inf f;
then (lim_inf f).x = lim_inf((R_EAL f)#x) by MESFUNC8:def 7;
hence thesis by Th1;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func lim_sup f -> PartFunc of X,ExtREAL equals
lim_sup R_EAL f;
coherence;
end;
theorem Th13:
for X be non empty set, f be Functional_Sequence of X,REAL holds
for x be Element of X st x in dom lim_sup f holds (lim_sup f).x = lim_sup R_EAL
(f#x)
proof
let X be non empty set, f be Functional_Sequence of X,REAL;
let x be Element of X;
assume x in dom lim_sup f;
then (lim_sup f).x = lim_sup((R_EAL f)#x) by MESFUNC8:def 8;
hence thesis by Th1;
end;
definition
let X be non empty set, f be Functional_Sequence of X,REAL;
func lim f -> PartFunc of X,ExtREAL equals
lim R_EAL f;
coherence;
end;
theorem Th14:
for X be non empty set, f be Functional_Sequence of X,REAL holds
for x be Element of X st x in dom lim f holds (lim f).x=lim R_EAL(f#x)
proof
let X be non empty set, f be Functional_Sequence of X,REAL;
let x be Element of X;
assume x in dom lim f;
then (lim f).x = lim((R_EAL f)#x) by MESFUNC8:def 9;
hence thesis by Th1;
end;
theorem Th15:
for f be Functional_Sequence of X,REAL, x be Element of X st x
in dom lim f & f#x is convergent holds (lim f).x= (lim_sup f).x & (lim f).x = (
lim_inf f).x
proof
let f be Functional_Sequence of X,REAL;
let x be Element of X;
assume that
A1: x in dom lim f and
A2: f#x is convergent;
R_EAL(f#x) is convergent by A2,RINFSUP2:14;
then
A3: lim R_EAL(f#x) = lim_sup R_EAL(f#x) & lim R_EAL(f#x) = lim_inf R_EAL(f#x
) by RINFSUP2:41;
A4: x in dom (f.0) by A1,MESFUNC8:def 9;
then x in dom lim_inf f by MESFUNC8:def 7;
then
A5: (lim_inf f).x = lim_inf R_EAL(f#x) by Th12;
x in dom lim_sup f by A4,MESFUNC8:def 8;
then (lim_sup f).x = lim_sup R_EAL(f#x) by Th13;
hence thesis by A1,A5,A3,Th14;
end;
theorem
for f be with_the_same_dom Functional_Sequence of X,REAL, F be
SetSequence of S, r be Real st (for n be Nat holds F.n = dom(
f.0) /\ great_dom(f.n,r)) holds union rng F = dom(f.0) /\ great_dom(sup f,r)
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, F be SetSequence
of S, r be Real;
set E = dom(f.0);
assume
A1: for n be Nat holds F.n = E /\ great_dom(f.n,r);
now
let x be object;
assume
A2: x in E /\ great_dom(sup f,r);
then reconsider z=x as Element of X;
A3: x in E by A2,XBOOLE_0:def 4;
x in great_dom(sup f,r) by A2,XBOOLE_0:def 4;
then
A4: r < (sup f).z by MESFUNC1:def 13;
ex n be Element of NAT st r < (f#z).n
proof
assume
A5: for n be Element of NAT holds (f#z).n <= r;
for p be ExtReal holds p in rng R_EAL(f#z) implies p <= r
proof
let p be ExtReal;
assume p in rng R_EAL(f#z);
then ex n be object st n in NAT & (R_EAL(f#z)).n = p by FUNCT_2:11;
hence thesis by A5;
end;
then r is UpperBound of rng R_EAL(f#z) by XXREAL_2:def 1;
then
A6: sup rng R_EAL(f#z) <= r by XXREAL_2:def 3;
x in dom sup f by A3,MESFUNC8:def 4;
hence contradiction by A4,A6,Th3;
end;
then consider n be Element of NAT such that
A7: r < (f#z).n;
A8: x in dom (f.n) by A3,MESFUNC8:def 2;
r < (f.n).z by A7,SEQFUNC:def 10;
then
A9: x in great_dom(f.n,r) by A8,MESFUNC1:def 13;
A10: F.n in rng F by FUNCT_2:4;
F.n = E /\ great_dom(f.n,r) by A1;
then x in F.n by A3,A9,XBOOLE_0:def 4;
hence x in union rng F by A10,TARSKI:def 4;
end;
then
A11: E /\ great_dom(sup f,r) c= union rng F;
now
let x be object;
assume x in union rng F;
then consider y be set such that
A12: x in y and
A13: y in rng(F qua SetSequence of X) by TARSKI:def 4;
reconsider z=x as Element of X by A12,A13;
consider n be object such that
A14: n in dom F and
A15: y=F.n by A13,FUNCT_1:def 3;
reconsider n as Element of NAT by A14;
A16: F.n = E /\ great_dom(f.n,r) by A1;
then x in great_dom(f.n,r) by A12,A15,XBOOLE_0:def 4;
then
A17: r < (f.n).z by MESFUNC1:def 13;
f#z = (R_EAL f)#z by Th1;
then (f.n).z = ((R_EAL f)#z).n by SEQFUNC:def 10;
then
A18: (f.n).z <= sup rng((R_EAL f)#z) by FUNCT_2:4,XXREAL_2:4;
A19: x in E by A12,A15,A16,XBOOLE_0:def 4;
then
A20: x in dom sup f by MESFUNC8:def 4;
then (sup f).z = sup((R_EALf)#z) by MESFUNC8:def 4;
then r < (sup f).z by A17,A18,XXREAL_0:2;
then x in great_dom(sup f,r) by A20,MESFUNC1:def 13;
hence x in E /\ great_dom(sup f,r) by A19,XBOOLE_0:def 4;
end;
then union rng F c= E /\ great_dom(sup f,r);
hence thesis by A11;
end;
theorem
for f be with_the_same_dom Functional_Sequence of X,REAL, F be
SetSequence of S, r be Real st (for n be Nat holds F.n = dom(
f.0) /\ great_eq_dom(f.n,r)) holds meet rng F = dom(f.0) /\ great_eq_dom(inf f,
r)
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, F be SetSequence
of S, r be Real;
set E = dom(f.0);
assume
A1: for n be Nat holds F.n = dom(f.0) /\ great_eq_dom(f.n,r);
now
let x be object;
assume
A2: x in meet rng(F qua SetSequence of X);
then reconsider z=x as Element of X;
A3: F.0 = E /\ great_eq_dom(f.0,r) by A1;
F.0 in rng F by FUNCT_2:4;
then x in F.0 by A2,SETFAM_1:def 1;
then
A4: x in E by A3,XBOOLE_0:def 4;
then
A5: x in dom inf f by MESFUNC8:def 3;
A6: now
let n be Element of NAT;
F.n in rng F by FUNCT_2:4;
then
A7: z in F.n by A2,SETFAM_1:def 1;
F.n = E /\ great_eq_dom(f.n,r) by A1;
then x in great_eq_dom(f.n,r) by A7,XBOOLE_0:def 4;
then r <= (f.n).z by MESFUNC1:def 14;
hence r <= (R_EAL(f#z)).n by SEQFUNC:def 10;
end;
for p be ExtReal holds p in rng R_EAL(f#z) implies r <= p
proof
let p be ExtReal;
assume p in rng R_EAL(f#z);
then ex n be object st n in NAT & (R_EAL(f#z)).n = p by FUNCT_2:11;
hence thesis by A6;
end;
then r is LowerBound of rng R_EAL(f#z) by XXREAL_2:def 2;
then r <= inf rng R_EAL(f#z) by XXREAL_2:def 4;
then r <= (inf f).x by A5,Th2;
then x in great_eq_dom(inf f,r) by A5,MESFUNC1:def 14;
hence x in E /\ great_eq_dom(inf f,r) by A4,XBOOLE_0:def 4;
end;
then
A8: meet rng F c= E /\ great_eq_dom(inf f,r);
now
let x be object;
assume
A9: x in E /\ great_eq_dom(inf f,r);
then reconsider z=x as Element of X;
A10: x in E by A9,XBOOLE_0:def 4;
x in great_eq_dom(inf f,r) by A9,XBOOLE_0:def 4;
then
A11: r <= (inf f).z by MESFUNC1:def 14;
now
let y be set;
assume y in rng F;
then consider n be object such that
A12: n in NAT and
A13: y=F.n by FUNCT_2:11;
reconsider n as Element of NAT by A12;
A14: x in dom (f.n) by A10,MESFUNC8:def 2;
x in dom inf f by A10,MESFUNC8:def 3;
then
A15: (inf f).z = inf rng R_EAL(f#z) by Th2;
(f.n).z = (R_EAL(f#z)).n by SEQFUNC:def 10;
then (f.n).z >= inf rng R_EAL(f#z) by FUNCT_2:4,XXREAL_2:3;
then r <= (f.n).z by A11,A15,XXREAL_0:2;
then
A16: x in great_eq_dom(f.n,r) by A14,MESFUNC1:def 14;
F.n = E /\ great_eq_dom(f.n,r) by A1;
hence x in y by A10,A13,A16,XBOOLE_0:def 4;
end;
hence x in meet rng F by SETFAM_1:def 1;
end;
then E /\ great_eq_dom(inf f,r) c= meet rng F;
hence thesis by A8;
end;
theorem Th18:
for f be with_the_same_dom Functional_Sequence of X,REAL, E be
Element of S st dom (f.0) = E & (for n be Nat holds f.n
is_measurable_on E) holds lim_sup f is_measurable_on E
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, E be Element of S;
assume that
A1: dom(f.0) = E and
A2: for n be Nat holds f.n is_measurable_on E;
for n being Nat holds (R_EAL f).n is_measurable_on E by A2,Th7;
hence thesis by A1,MESFUNC8:23;
end;
theorem
for f be with_the_same_dom Functional_Sequence of X,REAL, E be Element
of S st dom(f.0) = E & (for n be Nat holds f.n is_measurable_on E)
holds lim_inf f is_measurable_on E
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, E be Element of S;
assume that
A1: dom(f.0) = E and
A2: for n be Nat holds f.n is_measurable_on E;
for n being Nat holds (R_EAL f).n is_measurable_on E by A2,Th7;
hence thesis by A1,MESFUNC8:24;
end;
theorem
for f be Functional_Sequence of X,REAL, x be Element of X st x in dom
(f.0) & f#x is convergent holds (superior_realsequence f)#x is bounded_below
proof
let f be Functional_Sequence of X,REAL, x be Element of X;
assume
A1: x in dom (f.0);
assume f#x is convergent;
then
A2: f#x is bounded;
then superior_realsequence R_EAL(f#x) = superior_realsequence (f#x) by
RINFSUP2:9;
then
A3: (superior_realsequence f)#x = superior_realsequence(f#x) by A1,Th11;
superior_realsequence(f#x) is bounded by A2,RINFSUP1:56;
hence thesis by A3,RINFSUP2:13;
end;
theorem Th21:
for f be with_the_same_dom Functional_Sequence of X,REAL, E be
Element of S st dom(f.0) = E & (for n be Nat holds f.n
is_measurable_on E) & (for x be Element of X st x in E holds f#x is convergent)
holds lim f is_measurable_on E
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, E be Element of S;
assume
A1: dom (f.0) = E;
then
A2: dom lim f = E by MESFUNC8:def 9;
assume for n be Nat holds f.n is_measurable_on E;
then
A3: lim_sup f is_measurable_on E by A1,Th18;
assume
A4: for x be Element of X st x in E holds f#x is convergent;
A5: now
let x be Element of X;
assume
A6: x in dom lim f;
then f#x is convergent by A2,A4;
hence (lim f).x= (lim_sup f).x by A6,Th15;
end;
dom lim_sup f = E by A1,MESFUNC8:def 8;
hence thesis by A2,A3,A5,PARTFUN1:5;
end;
theorem Th22:
for f be with_the_same_dom Functional_Sequence of X,REAL, g be
PartFunc of X,ExtREAL, E be Element of S st dom(f.0) = E & (for n be Nat
holds f.n is_measurable_on E) & dom g = E & for x be Element of X st x
in E holds f#x is convergent & g.x = lim(f#x) holds g is_measurable_on E
proof
let f be with_the_same_dom Functional_Sequence of X,REAL, g be PartFunc of X
,ExtREAL, E be Element of S;
assume that
A1: dom (f.0) = E and
A2: for n be Nat holds f.n is_measurable_on E and
A3: dom g = E and
A4: for x be Element of X st x in E holds f#x is convergent & g.x = lim( f#x);
A5: dom lim f = E by A1,MESFUNC8:def 9;
now
let x be Element of X;
assume
A6: x in dom lim f;
then x in E by A1,MESFUNC8:def 9;
then f#x is convergent by A4;
then lim(f#x) = lim R_EAL(f#x) by RINFSUP2:14;
then g.x = lim R_EAL(f#x) by A4,A5,A6;
hence g.x = (lim f).x by A6,Th14;
end;
then
A7: g = lim f by A3,A5,PARTFUN1:5;
for x be Element of X st x in E holds f#x is convergent by A4;
hence thesis by A1,A2,A7,Th21;
end;
begin :: The Measurability of Complex-valued Functional Sequences
definition
let X be non empty set, H be Functional_Sequence of X,COMPLEX, x be Element
of X;
func H#x -> Complex_Sequence means
:Def9:
for n be Nat holds it.n = (H.n).x;
existence
proof
defpred P[Element of NAT,set] means $2 = (H.$1).x;
A1: for n being Element of NAT ex y being Element of COMPLEX st P[n,y]
proof
let n being Element of NAT;
(H.n).x in COMPLEX by XCMPLX_0:def 2;
hence thesis;
end;
consider f be sequence of COMPLEX such that
A2: for n be Element of NAT holds P[n,f.n] from FUNCT_2:sch 3(A1);
take f;
let n be Nat;
n in NAT by ORDINAL1:def 12;
hence thesis by A2;
end;
uniqueness
proof
let S1,S2 be Complex_Sequence such that
A3: for n be Nat holds S1.n = (H.n).x and
A4: for n be Nat holds S2.n = (H.n).x;
now
let n be Element of NAT;
S1.n = (H.n).x by A3;
hence S1.n = S2.n by A4;
end;
hence thesis by FUNCT_2:63;
end;
end;
definition
let X be non empty set, f be Functional_Sequence of X,COMPLEX;
func lim f -> PartFunc of X,COMPLEX means
:Def10:
dom it = dom (f.0) & for x
be Element of X st x in dom it holds it.x=lim(f#x);
existence
proof
defpred P[set] means $1 in dom(f.0);
deffunc F(Element of X) = In(lim (f#$1),COMPLEX);
consider g being PartFunc of X,COMPLEX such that
A1: (for x be Element of X holds x in dom g iff P[x]) & for x be
Element of X st x in dom g holds g/.x = F(x) from PARTFUN2:sch 2;
take g;
A2: now
let x be Element of X;
assume
A3: x in dom g;
then g/.x =F(x) by A1;
hence g.x =lim(f#x) by A3,PARTFUN1:def 6;
end;
for x be object holds x in dom g iff x in dom(f.0) by A1;
hence thesis by A2,TARSKI:2;
end;
uniqueness
proof
let g,h be PartFunc of X,COMPLEX;
assume that
A4: dom g = dom(f.0) and
A5: for x be Element of X st x in dom g holds g.x=lim(f#x);
assume that
A6: dom h = dom(f.0) and
A7: for x be Element of X st x in dom h holds h.x=lim(f#x);
now
let x be Element of X;
assume
A8: x in dom g;
then g.x = lim (f#x) by A5;
hence g.x = h.x by A4,A6,A7,A8;
end;
hence thesis by A4,A6,PARTFUN1:5;
end;
end;
definition
let X be non empty set;
let f be Functional_Sequence of X,COMPLEX;
func Re f -> Functional_Sequence of X,REAL means
:Def11:
for n be Nat
holds dom(it.n) = dom(f.n) & for x be Element of X st x in dom(it.n)
holds (it.n).x = (Re(f#x)).n;
existence
proof
defpred P[Element of NAT,Function] means dom $2 = dom(f.$1) & for x be
Element of X st x in dom $2 holds $2.x=(Re(f#x)).$1;
A1: for n being Element of NAT ex y being Element of PFuncs(X,REAL) st P[n, y]
proof
let n be Element of NAT;
deffunc F(Element of X) = In((Re(f#$1)).n,REAL);
defpred P[set] means $1 in dom(f.n);
consider g being PartFunc of X,REAL such that
A2: (for x be Element of X holds x in dom g iff P[x]) & for x be
Element of X st x in dom g holds g/.x = F(x) from PARTFUN2:sch 2;
take g;
A3: now
let x be Element of X;
assume
A4: x in dom g;
then g/.x = F(x) by A2
.=(Re(f#x)).n;
hence g.x =(Re(f#x)).n by A4,PARTFUN1:def 6;
end;
for x be object holds x in dom g iff x in dom(f.n) by A2;
hence thesis by A3,PARTFUN1:45,TARSKI:2;
end;
consider g being sequence of PFuncs(X,REAL) such that
A5: for n being Element of NAT holds P[n,g.n] from FUNCT_2:sch 3(A1);
reconsider g as Functional_Sequence of X,REAL;
take g;
thus for n holds dom(g.n) = dom(f.n) & for x be Element of X st x in dom(g
.n) holds (g.n).x = (Re(f#x)).n
proof
let n;
n in NAT by ORDINAL1:def 12;
hence thesis by A5;
end;
end;
uniqueness
proof
let g,h be Functional_Sequence of X,REAL;
assume
A6: for n holds dom (g.n) = dom (f.n) & for x be Element of X st x in
dom (g.n) holds (g.n).x=(Re(f#x)).n;
assume
A7: for n holds dom (h.n) = dom (f.n) & for x be Element of X st x in
dom (h.n) holds (h.n).x=(Re(f#x)).n;
now
let n be Element of NAT;
A8: dom(g.n) = dom(f.n) by A6
.=dom(h.n) by A7;
now
let x be Element of X;
assume
A9: x in dom (g.n);
then (g.n).x =(Re(f#x)).n by A6;
hence (g.n).x =(h.n).x by A7,A8,A9;
end;
hence g.n=h.n by A8,PARTFUN1:5;
end;
hence thesis by FUNCT_2:63;
end;
end;
registration
let X be non empty set;
let f be with_the_same_dom Functional_Sequence of X,COMPLEX;
cluster Re f -> with_the_same_dom;
correctness
proof
now
let k,l be Nat;
dom((Re f).k) = dom(f.k) by Def11;
then dom((Re f).k) = dom(f.l) by MESFUNC8:def 2;
hence dom((Re f).k) = dom((Re f).l) by Def11;
end;
hence thesis by MESFUNC8:def 2;
end;
end;
definition
let X be non empty set;
let f be Functional_Sequence of X,COMPLEX;
func Im f -> Functional_Sequence of X,REAL means
:Def12:
for n be Nat holds
dom(it.n) = dom(f.n) & for x be Element of X st x in dom(it.n)
holds (it.n).x = (Im(f#x)).n;
existence
proof
defpred P[Element of NAT,Function] means dom $2 = dom(f.$1) & for x be
Element of X st x in dom $2 holds $2.x=(Im(f#x)).$1;
A1: for n being Element of NAT ex y being Element of PFuncs(X,REAL) st P[n, y]
proof
let n be Element of NAT;
deffunc F(Element of X) = In((Im(f#$1)).n,REAL);
defpred P[set] means $1 in dom(f.n);
consider g being PartFunc of X,REAL such that
A2: (for x be Element of X holds x in dom g iff P[x]) & for x be
Element of X st x in dom g holds g/.x = F(x) from PARTFUN2:sch 2;
take g;
A3: now
let x be Element of X;
assume
A4: x in dom g;
then g/.x = F(x) by A2
.=(Im(f#x)).n;
hence g.x =(Im(f#x)).n by A4,PARTFUN1:def 6;
end;
for x be object holds x in dom g iff x in dom(f.n) by A2;
hence thesis by A3,PARTFUN1:45,TARSKI:2;
end;
consider g being sequence of PFuncs(X,REAL) such that
A5: for n being Element of NAT holds P[n,g.n] from FUNCT_2:sch 3(A1);
reconsider g as Functional_Sequence of X,REAL;
take g;
thus for n holds dom(g.n) = dom(f.n) & for x be Element of X st x in dom(g
.n) holds (g.n).x = (Im(f#x)).n
proof
let n;
n in NAT by ORDINAL1:def 12;
hence thesis by A5;
end;
end;
uniqueness
proof
let g,h be Functional_Sequence of X,REAL;
assume
A6: for n holds dom (g.n) = dom (f.n) & for x be Element of X st x in
dom (g.n) holds (g.n).x=(Im(f#x)).n;
assume
A7: for n holds dom (h.n) = dom (f.n) & for x be Element of X st x in
dom (h.n) holds (h.n).x=(Im(f#x)).n;
now
let n be Element of NAT;
A8: dom(g.n) = dom(f.n) by A6
.=dom(h.n) by A7;
now
let x be Element of X;
assume
A9: x in dom (g.n);
then (g.n).x = (Im(f#x)).n by A6;
hence (g.n).x =(h.n).x by A7,A8,A9;
end;
hence g.n=h.n by A8,PARTFUN1:5;
end;
hence thesis by FUNCT_2:63;
end;
end;
registration
let X be non empty set;
let f be with_the_same_dom Functional_Sequence of X,COMPLEX;
cluster Im f -> with_the_same_dom;
correctness
proof
now
let k,l be Nat;
dom((Im f).k) = dom(f.k) by Def12;
then dom((Im f).k) = dom(f.l) by MESFUNC8:def 2;
hence dom((Im f).k) = dom((Im f).l) by Def12;
end;
hence thesis by MESFUNC8:def 2;
end;
end;
theorem Th23:
for f be with_the_same_dom Functional_Sequence of X,COMPLEX, x
be Element of X st x in dom (f.0) holds (Re f)#x = Re(f#x) & (Im f)#x = Im(f#x)
proof
let f be with_the_same_dom Functional_Sequence of X,COMPLEX;
let x be Element of X;
set F = Re f;
set G = Im f;
assume
A1: x in dom (f.0);
now
let n be Element of NAT;
dom(F.n) = dom(f.n) by Def11;
then
A2: dom(F.n) = dom (f.0) by MESFUNC8:def 2;
dom(G.n) = dom(f.n) by Def12;
then
A3: dom(G.n) = dom (f.0) by MESFUNC8:def 2;
(F#x).n = (F.n).x & (G#x).n = (G.n).x by SEQFUNC:def 10;
hence (F#x).n = (Re(f#x)).n & (G#x).n = (Im(f#x)).n by A1,A2,A3,Def11,Def12
;
end;
hence thesis by FUNCT_2:63;
end;
theorem Th24:
for f be Functional_Sequence of X,COMPLEX, n be Nat
holds (Re f).n = Re(f.n) & (Im f).n = Im(f.n)
proof
let f be Functional_Sequence of X,COMPLEX;
let n be Nat;
dom((Re f).n) = dom(f.n) by Def11;
then
A1: dom((Re f).n) = dom(Re(f.n)) by COMSEQ_3:def 3;
now
let x be Element of X;
assume
A2: x in dom((Re f).n);
then Re(f.n).x = Re((f.n).x) by A1,COMSEQ_3:def 3;
then
A3: Re(f.n).x = Re((f#x).n) by Def9;
((Re f).n).x = (Re(f#x)).n by A2,Def11;
hence ((Re f).n).x = Re(f.n).x by A3,COMSEQ_3:def 5;
end;
hence (Re f).n = Re(f.n) by A1,PARTFUN1:5;
dom((Im f).n) = dom(f.n) by Def12;
then
A4: dom((Im f).n) = dom(Im(f.n)) by COMSEQ_3:def 4;
now
let x be Element of X;
assume
A5: x in dom((Im f).n);
then Im(f.n).x = Im((f.n).x) by A4,COMSEQ_3:def 4;
then
A6: Im(f.n).x = Im((f#x).n) by Def9;
((Im f).n).x = (Im(f#x)).n by A5,Def12;
hence ((Im f).n).x = Im(f.n).x by A6,COMSEQ_3:def 6;
end;
hence thesis by A4,PARTFUN1:5;
end;
theorem Th25:
for f be with_the_same_dom Functional_Sequence of X,COMPLEX st (
for x be Element of X st x in dom(f.0) holds f#x is convergent) holds lim Re f
= Re lim f & lim Im f = Im lim f
proof
let f be with_the_same_dom Functional_Sequence of X,COMPLEX;
dom lim Re f = dom((Re f).0) by MESFUNC8:def 9;
then
A1: dom lim Re f = dom(f.0) by Def11;
A2: dom Re lim f = dom lim f by COMSEQ_3:def 3;
then
A3: dom lim Re f = dom Re lim f by A1,Def10;
assume
A4: for x be Element of X st x in dom(f.0) holds f#x is convergent;
A5: now
let x be Element of X;
assume
A6: x in dom lim Re f;
then
A7: f#x is convergent by A4,A1;
then Re(f#x) is convergent;
then
A8: (Re f)#x is convergent by A1,A6,Th23;
(lim Re f).x = lim R_EAL((Re f)#x) by A6,Th14
.= lim((Re f)#x) by A8,RINFSUP2:14;
then (lim Re f).x = lim(Re(f#x)) by A1,A6,Th23;
then
A9: (lim Re f).x = Re lim(f#x) by A7,COMSEQ_3:41;
(Re lim f).x = Re((lim f).x) by A3,A6,COMSEQ_3:def 3;
hence (lim Re f).x = (Re lim f).x by A2,A3,A6,A9,Def10;
end;
Re lim f is PartFunc of X,ExtREAL by NUMBERS:31,RELSET_1:7;
hence lim Re f = Re lim f by A3,A5,PARTFUN1:5;
dom lim Im f = dom((Im f).0) by MESFUNC8:def 9;
then
A10: dom lim Im f = dom(f.0) by Def12;
A11: dom Im lim f = dom lim f by COMSEQ_3:def 4;
then
A12: dom lim Im f = dom Im lim f by A10,Def10;
A13: now
let x be Element of X;
assume
A14: x in dom lim Im f;
then
A15: f#x is convergent by A4,A10;
then Im(f#x) is convergent;
then
A16: (Im f)#x is convergent by A10,A14,Th23;
(lim Im f).x = lim R_EAL((Im f)#x) by A14,Th14
.= lim((Im f)#x) by A16,RINFSUP2:14;
then (lim Im f).x = lim Im(f#x) by A10,A14,Th23;
then
A17: (lim Im f).x = Im lim(f#x) by A15,COMSEQ_3:41;
(Im lim f).x = Im((lim f).x) by A12,A14,COMSEQ_3:def 4;
hence (lim Im f).x = (Im lim f).x by A11,A12,A14,A17,Def10;
end;
Im lim f is PartFunc of X,ExtREAL by NUMBERS:31,RELSET_1:7;
hence thesis by A12,A13,PARTFUN1:5;
end;
theorem
for f be with_the_same_dom Functional_Sequence of X,COMPLEX, E be
Element of S st dom(f.0) = E & (for n be Nat holds f.n
is_measurable_on E) & (for x be Element of X st x in E holds f#x is convergent)
holds lim f is_measurable_on E
proof
let f be with_the_same_dom Functional_Sequence of X,COMPLEX, E be Element of
S;
assume that
A1: dom (f.0) = E and
A2: for n be Nat holds f.n is_measurable_on E and
A3: for x be Element of X st x in E holds f#x is convergent;
A4: lim Im f = R_EAL Im lim f by A1,A3,Th25;
A5: now
let x be Element of X;
assume
A6: x in E;
then f#x is convergent by A3;
then Im(f#x) is convergent;
hence (Im f)#x is convergent by A1,A6,Th23;
end;
A7: now
let n be Nat;
f.n is_measurable_on E by A2;
then Im(f.n) is_measurable_on E by MESFUN6C:def 1;
hence (Im f).n is_measurable_on E by Th24;
end;
dom((Im f).0) = E by A1,Def12;
then lim Im f is_measurable_on E by A7,A5,Th21;
then
A8: Im lim f is_measurable_on E by A4,MESFUNC6:def 1;
A9: now
let x be Element of X;
assume
A10: x in E;
then f#x is convergent by A3;
then Re(f#x) is convergent;
hence (Re f)#x is convergent by A1,A10,Th23;
end;
A11: now
let n be Nat;
f.n is_measurable_on E by A2;
then Re(f.n) is_measurable_on E by MESFUN6C:def 1;
hence (Re f).n is_measurable_on E by Th24;
end;
A12: lim Re f = R_EAL Re lim f by A1,A3,Th25;
dom((Re f).0) = E by A1,Def11;
then lim Re f is_measurable_on E by A11,A9,Th21;
then Re lim f is_measurable_on E by A12,MESFUNC6:def 1;
hence thesis by A8,MESFUN6C:def 1;
end;
theorem
for f be with_the_same_dom Functional_Sequence of X,COMPLEX, g be
PartFunc of X,COMPLEX, E be Element of S st dom(f.0) = E & (for n be Nat
holds f.n is_measurable_on E) & dom g = E & for x be Element of X st x
in E holds f#x is convergent & g.x = lim(f#x) holds g is_measurable_on E
proof
let f be with_the_same_dom Functional_Sequence of X,COMPLEX, g be PartFunc
of X,COMPLEX, E be Element of S;
assume that
A1: dom (f.0) = E and
A2: for n be Nat holds f.n is_measurable_on E and
A3: dom g = E and
A4: for x be Element of X st x in E holds f#x is convergent & g.x = lim( f#x);
A5: now
let n be Nat;
f.n is_measurable_on E by A2;
then Im(f.n) is_measurable_on E by MESFUN6C:def 1;
hence (Im f).n is_measurable_on E by Th24;
end;
A6: dom Im g = E by A3,COMSEQ_3:def 4;
A7: now
let x be Element of X;
assume
A8: x in E;
then
A9: f#x is convergent by A4;
then Im(f#x) is convergent;
hence (Im f)#x is convergent by A1,A8,Th23;
g.x = lim(f#x) by A4,A8;
then Im(g.x) = lim Im(f#x) by A9,COMSEQ_3:41;
then Im(g.x) = lim((Im f)#x) by A1,A8,Th23;
hence (Im g).x = lim((Im f)#x) by A6,A8,COMSEQ_3:def 4;
end;
dom((Im f).0) = E by A1,Def12;
then R_EAL Im g is_measurable_on E by A5,A6,A7,Th22;
then
A10: Im g is_measurable_on E by MESFUNC6:def 1;
A11: now
let n be Nat;
f.n is_measurable_on E by A2;
then Re(f.n) is_measurable_on E by MESFUN6C:def 1;
hence (Re f).n is_measurable_on E by Th24;
end;
A12: dom Re g = E by A3,COMSEQ_3:def 3;
A13: now
let x be Element of X;
assume
A14: x in E;
then
A15: f#x is convergent by A4;
then Re(f#x) is convergent;
hence (Re f)#x is convergent by A1,A14,Th23;
g.x = lim(f#x) by A4,A14;
then Re(g.x) = lim Re(f#x) by A15,COMSEQ_3:41;
then Re(g.x) = lim((Re f)#x) by A1,A14,Th23;
hence (Re g).x = lim((Re f)#x) by A12,A14,COMSEQ_3:def 3;
end;
dom((Re f).0) = E by A1,Def11;
then R_EAL Re g is_measurable_on E by A11,A12,A13,Th22;
then Re g is_measurable_on E by MESFUNC6:def 1;
hence thesis by A10,MESFUN6C:def 1;
end;
begin :: Selected Properties of Complex-valued Measurable Functions
theorem
(r(#)f)|Y = r(#)(f|Y)
proof
A1: dom ((r(#)f)|Y) = dom (r(#)f) /\ Y by RELAT_1:61;
then dom ((r(#)f)|Y) = dom f /\ Y by VALUED_1:def 5;
then
A2: dom ((r(#)f)|Y) = dom (f|Y) by RELAT_1:61;
then
A3: dom ((r(#)f)|Y) = dom (r(#)(f|Y)) by VALUED_1:def 5;
now
let x be Element of X;
assume
A4: x in dom ((r(#)f)|Y);
then
A5: x in dom (r(#)f) by A1,XBOOLE_0:def 4;
thus ((r(#)f)|Y).x = (r(#)f).x by A4,FUNCT_1:47
.= r*(f.x) by A5,VALUED_1:def 5
.= r*((f|Y).x) by A2,A4,FUNCT_1:47
.= (r(#)(f|Y)).x by A3,A4,VALUED_1:def 5;
end;
hence thesis by A3,PARTFUN1:5;
end;
Lm1: |.f.| is nonnegative
proof
now
let x be object;
assume x in dom |.f.|;
then |.f.|.x = |.f.x.| by VALUED_1:def 11;
hence 0 <= |.f.|.x by COMPLEX1:46;
end;
hence thesis by MESFUNC6:52;
end;
theorem
0 <= k & E c= dom f & f is_measurable_on E implies |.f.| to_power k
is_measurable_on E
proof
assume that
A1: 0 <= k and
A2: E c= dom f and
A3: f is_measurable_on E;
A4: |.f.| is nonnegative by Lm1;
E c= dom |.f.| by A2,VALUED_1:def 11;
hence thesis by A1,A2,A3,A4,MESFUN6C:29,30;
end;
theorem Th30:
for f,g be PartFunc of X,REAL holds (R_EAL f)(#)(R_EAL g) = R_EAL(f(#)g)
proof
let f,g be PartFunc of X,REAL;
A1: dom ((R_EAL f)(#)(R_EAL g)) = dom(R_EAL f) /\ dom(R_EAL g) by
MESFUNC1:def 5;
A2: now
let x be Element of X;
assume
A3: x in dom((R_EAL f)(#)(R_EAL g));
then x in dom(f(#)g) by A1,VALUED_1:def 4;
then
A4: (f(#)g).x = f.x * g.x by VALUED_1:def 4;
((R_EAL f)(#)(R_EAL g)).x = (R_EAL f).x * (R_EAL g).x by A3,MESFUNC1:def 5;
hence ((R_EAL f)(#)(R_EAL g)).x = (R_EAL(f(#)g)).x by A4;
end;
dom ((R_EAL f)(#)(R_EAL g)) = dom R_EAL(f(#)g) by A1,VALUED_1:def 4;
hence thesis by A2,PARTFUN1:5;
end;
theorem Th31:
for f,g be PartFunc of X,REAL st dom f /\ dom g = E & f
is_measurable_on E & g is_measurable_on E holds f(#)g is_measurable_on E
proof
let f,g be PartFunc of X,REAL;
assume that
A1: dom f /\ dom g = E and
A2: f is_measurable_on E & g is_measurable_on E;
R_EAL f is_measurable_on E & R_EAL g is_measurable_on E by A2,MESFUNC6:def 1;
then (R_EAL f)(#)(R_EAL g) is_measurable_on E by A1,MESFUNC7:15;
then R_EAL(f(#)g) is_measurable_on E by Th30;
hence thesis by MESFUNC6:def 1;
end;
theorem Th32:
Re(f(#)g) = Re(f)(#)Re(g) - Im(f)(#)Im(g) & Im(f(#)g) = Im(f)(#)
Re(g) + Re(f)(#)Im(g)
proof
A1: dom(Re(f)(#)Re(g)) = dom Re(f) /\ dom Re(g) by VALUED_1:def 4;
A2: dom(Im(f)(#)Im(g)) = dom Im(f) /\ dom Im(g) by VALUED_1:def 4;
A3: dom Re f = dom f by COMSEQ_3:def 3;
A4: dom Im g = dom g by COMSEQ_3:def 4;
A5: dom Re g = dom g by COMSEQ_3:def 3;
A6: dom(Re(f(#)g)) = dom(f(#)g) by COMSEQ_3:def 3;
then
A7: dom(Re(f(#)g)) = dom f /\ dom g by VALUED_1:def 4;
A8: dom Im f = dom f by COMSEQ_3:def 4;
A9: dom(Re(f)(#)Re(g) - Im(f)(#)Im(g)) = dom(Re(f)(#)Re(g)) /\ dom(Im(f)(#)Im
(g)) by VALUED_1:12;
now
let x be object;
assume
A10: x in dom(Re(f(#)g));
then Re(f(#)g).x = Re((f(#)g).x) by COMSEQ_3:def 3;
then Re(f(#)g).x = Re(f.x * g.x) by A6,A10,VALUED_1:def 4;
then
A11: Re(f(#)g).x = Re(f.x) * Re(g.x) - Im(f.x) * Im(g.x) by COMPLEX1:9;
x in dom g by A7,A10,XBOOLE_0:def 4;
then
A12: (Re g).x = Re(g.x) & (Im g).x = Im(g.x) by A5,A4,COMSEQ_3:def 3,def 4;
x in dom f by A7,A10,XBOOLE_0:def 4;
then (Re f).x = Re(f.x) & (Im f).x = Im(f.x) by A3,A8,COMSEQ_3:def 3,def 4;
then
Re(f(#)g).x = ((Re f)(#)(Re g)).x - (Im f).x * (Im g).x by A7,A1,A3,A5,A10
,A11,A12,VALUED_1:def 4;
then
Re(f(#)g).x = ((Re f)(#)(Re g)).x - ((Im f)(#)(Im g)).x by A7,A2,A8,A4,A10,
VALUED_1:def 4;
hence
Re(f(#)g).x = ( Re(f)(#)Re(g) - Im(f)(#)Im(g) ).x by A7,A9,A1,A2,A3,A8,A5
,A4,A10,VALUED_1:13;
end;
hence Re(f(#)g) = Re(f)(#)Re(g) - Im(f)(#)Im(g) by A7,A9,A1,A2,A3,A8,A5,A4,
FUNCT_1:2;
A13: dom(Im(f)(#)Re(g)) = dom Im(f) /\ dom Re(g) by VALUED_1:def 4;
A14: dom(Re(f)(#)Im(g)) = dom Re(f) /\ dom Im(g) by VALUED_1:def 4;
A15: dom(Im(f(#)g)) = dom(f(#)g) by COMSEQ_3:def 4;
then
A16: dom(Im(f(#)g)) = dom f /\ dom g by VALUED_1:def 4;
A17: dom(Im(f)(#)Re(g) + Re(f)(#)Im(g)) = dom(Im(f)(#)Re(g)) /\ dom(Re(f)(#)Im
(g)) by VALUED_1:def 1;
now
let x be object;
assume
A18: x in dom(Im(f(#)g));
then Im(f(#)g).x = Im((f(#)g).x) by COMSEQ_3:def 4;
then Im(f(#)g).x = Im(f.x * g.x) by A15,A18,VALUED_1:def 4;
then
A19: Im(f(#)g).x = Im(f.x) * Re(g.x) + Re(f.x) * Im(g.x) by COMPLEX1:9;
x in dom g by A16,A18,XBOOLE_0:def 4;
then
A20: Re(g).x = Re(g.x) & Im(g).x = Im(g.x) by A5,A4,COMSEQ_3:def 3,def 4;
x in dom f by A16,A18,XBOOLE_0:def 4;
then Re(f).x = Re(f.x) & Im(f).x = Im(f.x) by A3,A8,COMSEQ_3:def 3,def 4;
then
Im(f(#)g).x = ((Im f)(#)(Re g)).x + (Re f).x * (Im g).x by A16,A13,A8,A5
,A18,A19,A20,VALUED_1:def 4;
then
Im(f(#)g).x = ((Im f)(#)(Re g)).x + ((Re f)(#)(Im g)).x by A16,A14,A3,A4
,A18,VALUED_1:def 4;
hence
Im(f(#)g).x = ( Im(f)(#)Re(g) + Re(f)(#)Im(g) ).x by A16,A17,A13,A14,A3,A8
,A5,A4,A18,VALUED_1:def 1;
end;
hence thesis by A16,A17,A13,A14,A3,A8,A5,A4,FUNCT_1:2;
end;
theorem
dom f /\ dom g = E & f is_measurable_on E & g is_measurable_on E
implies f(#)g is_measurable_on E
proof
assume that
A1: dom f /\ dom g = E and
A2: f is_measurable_on E and
A3: g is_measurable_on E;
A4: dom Im g = dom g by COMSEQ_3:def 4;
A5: Im f is_measurable_on E by A2,MESFUN6C:def 1;
A6: dom Im f = dom f by COMSEQ_3:def 4;
then
A7: dom(Im(f)(#)Im(g)) = E by A1,A4,VALUED_1:def 4;
A8: Im g is_measurable_on E by A3,MESFUN6C:def 1;
then
A9: Im(f)(#)Im(g) is_measurable_on E by A1,A5,A6,A4,Th31;
A10: dom Re f = dom f by COMSEQ_3:def 3;
A11: dom Re g = dom g by COMSEQ_3:def 3;
A12: Re g is_measurable_on E by A3,MESFUN6C:def 1;
then
A13: Im(f)(#)Re(g) is_measurable_on E by A1,A5,A6,A11,Th31;
A14: Re f is_measurable_on E by A2,MESFUN6C:def 1;
then Re(f)(#)Im(g) is_measurable_on E by A1,A8,A10,A4,Th31;
then Im(f)(#)Re(g) + Re(f)(#)Im(g) is_measurable_on E by A13,MESFUNC6:26;
then
A15: Im(f(#)g) is_measurable_on E by Th32;
Re(f)(#)Re(g) is_measurable_on E by A1,A14,A12,A10,A11,Th31;
then Re(f)(#)Re(g) - Im(f)(#)Im(g) is_measurable_on E by A9,A7,MESFUNC6:29;
then Re(f(#)g) is_measurable_on E by Th32;
hence thesis by A15,MESFUN6C:def 1;
end;
theorem Th34:
for f,g be PartFunc of X,REAL st (ex E be Element of S st E =
dom f & E= dom g & f is_measurable_on E & g is_measurable_on E) & f is
nonnegative & g is nonnegative & (for x be Element of X st x in dom g holds g.x
<= f.x) holds Integral(M,g) <= Integral(M,f)
proof
let f,g be PartFunc of X,REAL;
assume that
A1: ex A be Element of S st A = dom f & A= dom g & f is_measurable_on A &
g is_measurable_on A and
A2: f is nonnegative & g is nonnegative and
A3: for x be Element of X st x in dom g holds g.x <= f.x;
A4: Integral(M,g) = integral+(M,R_EAL g) & Integral(M,f) = integral+(M,R_EAL
f) by A1,A2,MESFUNC6:82;
consider A be Element of S such that
A5: A = dom f & A= dom g and
A6: f is_measurable_on A & g is_measurable_on A by A1;
R_EAL f is_measurable_on A & R_EAL g is_measurable_on A by A6,MESFUNC6:def 1;
hence thesis by A2,A3,A5,A4,MESFUNC5:85;
end;
theorem Th35: :: MESFUN6C:31
for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, f be PartFunc of X,COMPLEX st f is_integrable_on M holds (ex A be Element
of S st A = dom f & f is_measurable_on A) & |.f.| is_integrable_on M
proof
let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
PartFunc of X,COMPLEX;
assume
A1: f is_integrable_on M;
then Re f is_integrable_on M by MESFUN6C:def 2;
then R_EAL Re f is_integrable_on M by MESFUNC6:def 4;
then consider A1 be Element of S such that
A2: A1 = dom R_EAL Re f and
A3: R_EAL Re f is_measurable_on A1;
A4: Re f is_measurable_on A1 by A3,MESFUNC6:def 1;
Im f is_integrable_on M by A1,MESFUN6C:def 2;
then R_EAL Im f is_integrable_on M by MESFUNC6:def 4;
then consider A2 be Element of S such that
A5: A2 = dom R_EAL Im f and
A6: R_EAL Im f is_measurable_on A2;
A7: A1 = dom f by A2,COMSEQ_3:def 3;
A2 = dom f by A5,COMSEQ_3:def 4;
then Im f is_measurable_on A1 by A6,A7,MESFUNC6:def 1;
then
A8: f is_measurable_on A1 by A4,MESFUN6C:def 1;
hence ex A be Element of S st A = dom f & f is_measurable_on A by A7;
thus thesis by A1,A7,A8,MESFUN6C:31;
end;
theorem
f is_integrable_on M implies ex F be sequence of S st (for n be
Nat holds F.n = dom f /\ great_eq_dom(|.f.|, (1/(n+1)))) & dom f \ eq_dom(
|.f.|,0) = union rng F & for n be Nat holds F.n in S & M.(F.n) <+infty
proof
assume
A1: f is_integrable_on M;
then consider E be Element of S such that
A2: E = dom f and
A3: f is_measurable_on E by Th35;
defpred P[Element of NAT,set] means $2 = E /\ great_eq_dom(|.f.|,1/($1+1));
A4: dom |.f.| = E by A2,VALUED_1:def 11;
now
let x be object;
reconsider y=|.f.|.x as Real;
assume
A5: x in E \ eq_dom(|.f.|, 0);
then
A6: x in E by XBOOLE_0:def 5;
then
A7: x in dom |.f.| by A2,VALUED_1:def 11;
not x in eq_dom(|.f.|, 0) by A5,XBOOLE_0:def 5;
then not y = 0 by A7,MESFUNC6:7;
then not |.f.x.| = 0 by A7,VALUED_1:def 11;
then f.x <> 0 by COMPLEX1:5,SQUARE_1:17;
then 0 < |.f.x.| by COMPLEX1:47;
then 0 < (|.f.|).x by A7,VALUED_1:def 11;
then x in great_dom(|.f.|, 0) by A7,MESFUNC1:def 13;
hence x in E /\ great_dom(|.f.|, 0) by A6,XBOOLE_0:def 4;
end;
then
A8: E \ eq_dom(|.f.|, 0) c= E /\ great_dom(|.f.|, 0);
now
let x be object;
assume
A9: x in E /\ great_dom(|.f.|, 0);
then x in great_dom(|.f.|, 0) by XBOOLE_0:def 4;
then 0 < (|.f.|).x by MESFUNC1:def 13;
then
A10: not x in eq_dom(|.f.|, 0) by MESFUNC1:def 15;
x in E by A9,XBOOLE_0:def 4;
hence x in E \ eq_dom(|.f.|, 0) by A10,XBOOLE_0:def 5;
end;
then
A11: E /\ great_dom(|.f.|, 0) c= E \ eq_dom(|.f.|, 0);
A12: |.f.| is_measurable_on E by A2,A3,MESFUN6C:30;
A13: for n be Element of NAT ex Z be Element of S st P[n,Z]
proof
let n be Element of NAT;
take E /\ great_eq_dom(|.f.|,1/(n+1));
thus thesis by A12,A4,MESFUNC6:13;
end;
consider F be sequence of S such that
A14: for n be Element of NAT holds P[n,F.n] from FUNCT_2:sch 3(A13);
A15: for n be Nat holds F.n in S & M.(F.n) <+infty
proof
|.f.| is_integrable_on M by A1,Th35;
then
A16: Integral(M,|.f.|) < +infty by MESFUNC6:90;
let n be Nat;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
set z = (1/(n+1));
A17: F.n1 = E /\ great_eq_dom(|.f.| ,1/(n1+1)) by A14;
then reconsider En=F.n as Element of S;
set h = (|.f.|)|En;
consider nf be PartFunc of X,REAL such that
A18: nf is_simple_func_in S and
A19: dom nf = En and
A20: for x be object st x in En holds nf.x = 1/(n+1) by MESFUNC6:75;
A21: dom h = En by A4,A17,RELAT_1:62,XBOOLE_1:17;
A22: F.n c= great_eq_dom(|.f.|, 1/(n+1)) by A17,XBOOLE_1:17;
A23: for x be Element of X st x in dom nf holds nf.x <= h.x
proof
let x be Element of X;
assume
A24: x in dom nf;
then h.x = |.f.| .x by A19,FUNCT_1:49;
then 1/(n+1) <= h.x by A22,A19,A24,MESFUNC1:def 14;
hence thesis by A19,A20,A24;
end;
dom |.f.| /\ En = E /\ En by A2,VALUED_1:def 11;
then
A25: dom |.f.| /\ En = En by A17,XBOOLE_1:17,28;
|.f.| is_measurable_on En by A12,A17,MESFUNC6:16,XBOOLE_1:17;
then
A26: h is_measurable_on En by A25,MESFUNC6:76;
A27: h is nonnegative by Lm1,MESFUNC6:55;
for x be object st x in dom nf holds nf.x >= 0 by A19,A20;
then
A28: nf is nonnegative by MESFUNC6:52;
|.f.| is nonnegative & (|.f.|)|E=|.f.| by A4,Lm1;
then
A29: Integral(M,h) <= Integral(M,|.f.|) by A12,A4,A17,MESFUNC6:87,XBOOLE_1:17;
nf is_measurable_on En by A18,MESFUNC6:50;
then Integral(M,nf) <= Integral(M,h) by A21,A26,A27,A19,A28,A23,Th34;
then
A30: Integral(M,nf) <= Integral(M,|.f.|) by A29,XXREAL_0:2;
A31: z* M.En / z = M.En & +infty / z = +infty by XXREAL_3:83,88;
Integral(M,nf) = (1/(n+1)) * M.En by A19,A20,MESFUNC6:97;
then (1/(n+1)) * M.En < +infty by A16,A30,XXREAL_0:2;
hence thesis by A31,XXREAL_3:80;
end;
take F;
A32: for n be Nat holds F.n = E /\ great_eq_dom(|.f.|, 1/(n+1))
proof
let n be Nat;
n in NAT by ORDINAL1:def 12;
hence thesis by A14;
end;
then for n be Nat holds F.n = E /\ great_eq_dom(|.f.|, 0 + 1/(n+1));
then E /\ great_dom(|.f.|, 0) = union rng F by MESFUNC6:11;
hence thesis by A2,A32,A11,A8,A15;
end;
reserve x,A for set;
theorem Th37:
(|.f.|)|A = |. f|A .|
proof
dom((|.f.|)|A) = dom |.f.| /\ A by RELAT_1:61;
then
A1: dom((|.f.|)|A) = dom f /\ A by VALUED_1:def 11;
A2: dom(f|A) = dom f /\ A by RELAT_1:61;
then
A3: dom|. f|A .| = dom f /\ A by VALUED_1:def 11;
now
let x be Element of X;
assume
A4: x in dom((|.f.|)|A);
then (|.(f|A).|).x = |. (f|A).x .| by A1,A3,VALUED_1:def 11;
then
A5: (|.(f|A).|).x = |. f.x .| by A2,A1,A4,FUNCT_1:47;
x in dom f by A1,A4,XBOOLE_0:def 4;
then
A6: x in dom |.f.| by VALUED_1:def 11;
((|.f.|)|A).x = (|.f.|).x by A4,FUNCT_1:47;
hence ((|.f.|)|A).x = (|. f|A .|).x by A6,A5,VALUED_1:def 11;
end;
hence thesis by A1,A3,PARTFUN1:5;
end;
theorem Th38:
dom(|.f.|+|.g.|) = dom f /\ dom g & dom |.f+g.| c= dom |.f.|
proof
dom(|.f.|+|.g.|) = dom |.f.| /\ dom |.g.| by VALUED_1:def 1;
then dom(|.f.|+|.g.|) = dom f /\ dom |.g.| by VALUED_1:def 11;
hence dom(|.f.|+|.g.|) = dom f /\ dom g by VALUED_1:def 11;
dom |.f+g.| = dom(f+g) by VALUED_1:def 11
.= dom f /\ dom g by VALUED_1:def 1;
then dom |.f+g.| c= dom f by XBOOLE_1:17;
hence thesis by VALUED_1:def 11;
end;
theorem Th39:
(|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|) = (|.f.|+|.g.|)|( dom |.f+g.|)
proof
A1: dom |.f+g.| c= dom |.g.| by Th38;
then
A2: dom |.f+g.| c= dom g by VALUED_1:def 11;
dom(g|(dom |.f+g.|)) = dom g /\ dom |.f+g.| by RELAT_1:61;
then
A3: dom(g|(dom |.f+g.|)) = dom |.f+g.| by A2,XBOOLE_1:28;
then
A4: dom |.(g|(dom |.f+g.|)).| = dom |.f+g.| by VALUED_1:def 11;
A5: dom |.f+g.| c= dom |.f.| by Th38;
then
A6: dom |.f+g.| c= dom f by VALUED_1:def 11;
then dom |.f+g.| /\ dom |.f+g.| c= dom f /\ dom g by A2,XBOOLE_1:27;
then
A7: dom |.f+g.| c= dom(|.f.|+|.g.|) by Th38;
then
A8: dom((|.f.|+|.g.|)|(dom |.f+g.|)) = dom |.f+g.| by RELAT_1:62;
dom(f|(dom |.f+g.|)) = dom f /\ dom |.f+g.| by RELAT_1:61;
then
A9: dom(f|(dom |.f+g.|)) = dom |.f+g.| by A6,XBOOLE_1:28;
A10: (|.f.|)|(dom |.f+g.|) = |.(f|(dom |.f+g.|)).| & (|.g.|)|(dom |.f+g.|) =
|.(g |(dom |.f+g.|)).| by Th37;
then
A11: dom((|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|)) = dom (f|(dom |.f+g.|
)) /\ dom (g|(dom |.f+g.|)) by Th38
.= dom |.f+g.| by A9,A3;
A12: dom |.(f|(dom |.f+g.|)).| = dom |.f+g.| by A9,VALUED_1:def 11;
now
let x be Element of X;
assume
A13: x in dom((|.f.|+|.g.|)|(dom |.f+g.|));
then
A14: ((|.f.|+|.g.|)|(dom |.f+g.|)).x = (|.f.|+|.g.|).x by FUNCT_1:47
.= (|.f.|).x + (|.g.|).x by A7,A8,A13,VALUED_1:def 1
.= |. f.x .| + (|.g.|).x by A5,A8,A13,VALUED_1:def 11;
A15: x in dom |.f+g.| by A7,A13,RELAT_1:62;
then
((|.f.|)|(dom |.f+g.|) + (|.g.|)|(dom |.f+g.|)).x = ((|.f.|)|(dom |.f
+g.|)).x + ((|.g.|)|(dom |.f+g.|)).x by A11,VALUED_1:def 1
.= |.(f|(dom |.f+g.|)).x .| + |.(g|(dom |.f+g.|)).|.x by A12,A10,A15,
VALUED_1:def 11
.= |.(f|(dom |.f+g.|)).x .| + |.(g|(dom |.f+g.|)).x .| by A4,A15,
VALUED_1:def 11
.= |. f.x .| + |.(g|(dom |.f+g.|)).x .| by A15,FUNCT_1:49
.= |. f.x .| + |. g.x .| by A15,FUNCT_1:49;
hence ((|.f.|+|.g.|)|(dom |.f+g.|)).x = ((|.f.|)|(dom |.f+g.|) + (|.g.|)|(
dom |.f+g.|)).x by A1,A8,A13,A14,VALUED_1:def 11;
end;
hence thesis by A11,A7,PARTFUN1:5,RELAT_1:62;
end;
theorem Th40:
x in dom |.f+g.| implies (|.f+g.|).x <= (|.f.|+|.g.|).x
proof
A1: |. f.x + g.x .| <= |. f.x .| + |. g.x .| by COMPLEX1:56;
assume
A2: x in dom |.f+g.|;
then x in dom (f+g) by VALUED_1:def 11;
then
A3: |. (f+g).x .| <= |. f.x .| + |. g.x .| by A1,VALUED_1:def 1;
A4: dom |.f+g.| c= dom |.g.| by Th38;
then
A5: |. g.x .| = |.g.| .x by A2,VALUED_1:def 11;
x in dom |.g.| by A2,A4;
then
A6: x in dom g by VALUED_1:def 11;
A7: dom |.f+g.| c= dom |.f.| by Th38;
then x in dom |.f.| by A2;
then x in dom f by VALUED_1:def 11;
then x in dom f /\ dom g by A6,XBOOLE_0:def 4;
then
A8: x in dom(|.f.| + |.g.|) by Th38;
|. f.x .| = |.f.| .x by A2,A7,VALUED_1:def 11;
then |. f.x .| + |. g.x .| = (|.f.| + |.g.|).x by A5,A8,VALUED_1:def 1;
hence thesis by A2,A3,VALUED_1:def 11;
end;
theorem Th41:
for f,g be PartFunc of X,REAL st (for x be set st x in dom f
holds f.x <= g.x) holds g-f is nonnegative
proof
let f,g be PartFunc of X,REAL;
assume
A1: for x be set st x in dom f holds f.x <= g.x;
now
let x be object;
assume
A2: x in dom(g-f);
then x in dom g /\ dom f by VALUED_1:12;
then x in dom f by XBOOLE_0:def 4;
then 0 <= g.x - f.x by A1,XREAL_1:48;
hence 0 <=(g-f).x by A2,VALUED_1:13;
end;
hence thesis by MESFUNC6:52;
end;
theorem
f is_integrable_on M & g is_integrable_on M implies ex E be Element of
S st E = dom(f+g) & Integral(M,(|.f+g.|)|E) <= Integral(M,(|.f.|)|E) + Integral
(M,(|.g.|)|E)
proof
assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
|.f.| is_integrable_on M & |.g.| is_integrable_on M by A1,A2,Th35;
then
A3: |.f.|+|.g.| is_integrable_on M by MESFUNC6:100;
Im g is_integrable_on M by A2,MESFUN6C:def 2;
then R_EAL Im g is_integrable_on M by MESFUNC6:def 4;
then consider B2 be Element of S such that
A4: B2 = dom R_EAL Im g & R_EAL Im g is_measurable_on B2;
Im f is_integrable_on M by A1,MESFUN6C:def 2;
then R_EAL Im f is_integrable_on M by MESFUNC6:def 4;
then consider A2 be Element of S such that
A5: A2 = dom R_EAL Im f & R_EAL Im f is_measurable_on A2;
Re g is_integrable_on M by A2,MESFUN6C:def 2;
then R_EAL Re g is_integrable_on M by MESFUNC6:def 4;
then consider B1 be Element of S such that
A6: B1 = dom R_EAL Re g and
A7: R_EAL Re g is_measurable_on B1;
A8: B1 = dom g by A6,COMSEQ_3:def 3;
f+g is_integrable_on M by A1,A2,MESFUN6C:33;
then
A9: |.f+g.| is_integrable_on M by Th35;
set G = |.g.|;
set F = |.f.|;
for x be set st x in dom |.f+g.| holds (|.f+g.|).x <= (|.f.|+|.g.|).x
by Th40;
then
A10: (|.f.|+|.g.|) - |.f+g.| is nonnegative by Th41;
Re f is_integrable_on M by A1,MESFUN6C:def 2;
then R_EAL Re f is_integrable_on M by MESFUNC6:def 4;
then consider A1 be Element of S such that
A11: A1 = dom R_EAL Re f and
A12: R_EAL Re f is_measurable_on A1;
A13: A1 = dom f by A11,COMSEQ_3:def 3;
reconsider A = A1 /\ B1 as Element of S;
Re f is_measurable_on A1 by A12,MESFUNC6:def 1;
then
A14: Re f is_measurable_on A by MESFUNC6:16,XBOOLE_1:17;
A15: dom(f+g) = dom f /\ dom g by VALUED_1:def 1;
then
A16: dom |.f+g.| = A by A13,A8,VALUED_1:def 11;
Re g is_measurable_on B1 by A7,MESFUNC6:def 1;
then
A17: Re g is_measurable_on A by MESFUNC6:16,XBOOLE_1:17;
B2 = dom g & Im g is_measurable_on B2 by A4,COMSEQ_3:def 4,MESFUNC6:def 1;
then Im g is_measurable_on A by A8,MESFUNC6:16,XBOOLE_1:17;
then
A18: g is_measurable_on A by A17,MESFUN6C:def 1;
then
A19: |.g.| is_measurable_on A by A8,MESFUN6C:30,XBOOLE_1:17;
A2 = dom f & Im f is_measurable_on A2 by A5,COMSEQ_3:def 4,MESFUNC6:def 1;
then Im f is_measurable_on A by A13,MESFUNC6:16,XBOOLE_1:17;
then
A20: f is_measurable_on A by A14,MESFUN6C:def 1;
then |.f.| is_measurable_on A by A13,MESFUN6C:30,XBOOLE_1:17;
then
A21: |.f.|+|.g.| is_measurable_on A by A19,MESFUNC6:26;
A c= A1 by XBOOLE_1:17;
then
A22: A c= dom |.f.| by A13,VALUED_1:def 11;
A c= B1 by XBOOLE_1:17;
then
A23: A c= dom |.g.| by A8,VALUED_1:def 11;
A24: dom(|.f.|+|.g.|) = dom |.f.| /\ dom |.g.| by VALUED_1:def 1;
then
A25: dom |.f+g.| /\ dom(|.f.|+|.g.|) = A by A22,A23,A16,XBOOLE_1:19,28;
|.f+g.| is_measurable_on A by A13,A8,A20,A18,A15,MESFUN6C:11,30;
then consider E be Element of S such that
A26: E = dom(|.f.|+|.g.|) /\ dom |.f+g.| and
A27: Integral(M,(|.f+g.|)|E) <= Integral(M,(|.f.|+|.g.|)|E) by A21,A3,A25,A9
,A10,MESFUN6C:42;
A28: dom(G|E) = E by A23,A25,A26,RELAT_1:62;
take E;
thus E = dom(f+g) by A13,A8,A15,A24,A22,A23,A16,A26,XBOOLE_1:19,28;
F|E is_integrable_on M & G|E is_integrable_on M by A1,A2,Th35,MESFUNC6:91;
then consider E1 be Element of S such that
A29: E1 = dom (F|E) /\ dom (G|E) and
A30: Integral(M,F|E+G|E) = Integral(M,(F|E)|E1) + Integral(M,(G|E)|E1)
by MESFUNC6:101;
dom(F|E) = E by A22,A25,A26,RELAT_1:62;
then (F|E)|E1 = F|E & (G|E)|E1 = G|E by A29,A28;
hence thesis by A16,A25,A26,A27,A30,Th39;
end;
begin :: Properties of Complex-valued Simple Functions
definition
let X be non empty set;
let S be SigmaField of X;
let f be PartFunc of X,COMPLEX;
pred f is_simple_func_in S means
ex F being Finite_Sep_Sequence of S
st dom f = union rng F & for n being Nat,x,y being Element of X st n in dom F &
x in F.n & y in F.n holds f.x = f.y;
end;
definition
let X be non empty set;
let S be SigmaField of X;
let f be PartFunc of X,REAL;
let F be Finite_Sep_Sequence of S;
let a be FinSequence of REAL;
pred F,a are_Re-presentation_of f means
dom f = union rng F & dom F
= dom a & for n be Nat st n in dom F for x be set st x in F.n holds f.x=a.n;
end;
definition
let X,S,f;
let F be Finite_Sep_Sequence of S;
let a be FinSequence of COMPLEX;
pred F,a are_Re-presentation_of f means
dom f = union rng F & dom F
= dom a & for n be Nat st n in dom F for x be set st x in F.n holds f.x=a.n;
end;
theorem
f is_simple_func_in S iff Re f is_simple_func_in S & Im f is_simple_func_in S
proof
hereby
assume f is_simple_func_in S;
then consider F be Finite_Sep_Sequence of S such that
A1: dom f = union rng F and
A2: for n being Nat,x,y being Element of X st n in dom F & x in F.n &
y in F.n holds f.x = f.y;
A3: dom Im f = union rng F by A1,COMSEQ_3:def 4;
A4: for n being Nat,x,y being Element of X st n in dom F & x in F.n & y
in F.n holds (Im f).x = (Im f).y
proof
let n be Nat,x,y be Element of X;
assume that
A5: n in dom F and
A6: x in F.n & y in F.n;
F.n c= union rng F by A5,MESFUNC3:7;
then (Im f).x = Im(f.x) & (Im f).y = Im(f.y) by A3,A6,COMSEQ_3:def 4;
hence thesis by A2,A5,A6;
end;
A7: dom Re f = union rng F by A1,COMSEQ_3:def 3;
for n being Nat,x,y being Element of X st n in dom F & x in F.n & y in
F.n holds (Re f).x = (Re f).y
proof
let n be Nat,x,y be Element of X;
assume that
A8: n in dom F and
A9: x in F.n & y in F.n;
F.n c= union rng F by A8,MESFUNC3:7;
then (Re f).x = Re(f.x) & (Re f).y = Re(f.y) by A7,A9,COMSEQ_3:def 3;
hence thesis by A2,A8,A9;
end;
hence Re f is_simple_func_in S & Im f is_simple_func_in S by A7,A3,A4,
MESFUNC6:def 2;
end;
assume that
A10: Re f is_simple_func_in S and
A11: Im f is_simple_func_in S;
consider F be Finite_Sep_Sequence of S such that
A12: dom Re f = union rng F and
A13: for n being Nat,x,y being Element of X st n in dom F & x in F.n & y
in F.n holds (Re f).x = (Re f).y by A10,MESFUNC6:def 2;
set la = len F;
A14: dom f = union rng F by A12,COMSEQ_3:def 3;
consider G be Finite_Sep_Sequence of S such that
A15: dom Im f = union rng G and
A16: for n being Nat,x,y being Element of X st n in dom G & x in G.n & y
in G.n holds (Im f).x = (Im f).y by A11,MESFUNC6:def 2;
set lb = len G;
deffunc FG1(Nat) = F.(($1-'1) div lb + 1) /\ G.(($1-'1) mod lb + 1);
consider FG be FinSequence such that
A17: len FG = la*lb and
A18: for k be Nat st k in dom FG holds FG.k=FG1(k) from FINSEQ_1:sch 2;
A19: dom FG = Seg(la*lb) by A17,FINSEQ_1:def 3;
now
let k be Nat;
A20: lb divides la*lb by NAT_D:def 3;
set j=(k-'1) mod lb + 1;
set i=(k-'1) div lb + 1;
assume
A21: k in dom FG;
then
A22: 1 <= k by FINSEQ_3:25;
then
A23: lb > 0 by A17,A21,FINSEQ_3:25;
A24: k <= la*lb by A17,A21,FINSEQ_3:25;
then k -' 1 <= la*lb -' 1 by NAT_D:42;
then
A25: (k -' 1) div lb <= (la*lb -' 1) div lb by NAT_2:24;
A26: 1 <= la*lb by A22,A24,XXREAL_0:2;
1 <= lb by A23,NAT_1:14;
then (la*lb -' 1) div lb = la*lb div lb - 1 by A26,A20,NAT_2:15;
then i <= la*lb div lb by A25,XREAL_1:19;
then i >= 1 & i <= la by A23,NAT_1:14,NAT_D:18;
then i in dom F by FINSEQ_3:25;
then
A27: F.i in rng F by FUNCT_1:3;
(k -' 1) mod lb < lb by A23,NAT_D:1;
then j >= 1 & j <= lb by NAT_1:13,14;
then j in dom G by FINSEQ_3:25;
then G.j in rng G by FUNCT_1:3;
then F.i /\ G.j in S by A27,MEASURE1:34;
hence FG.k in S by A18,A21;
end;
then reconsider FG as FinSequence of S by FINSEQ_2:12;
for k,l be Nat st k in dom FG & l in dom FG & k <> l holds FG.k misses FG.l
proof
let k,l be Nat;
assume that
A28: k in dom FG and
A29: l in dom FG and
A30: k <> l;
set m=(l-'1) mod lb + 1;
set n=(l-'1) div lb + 1;
A31: 1 <= k by A28,FINSEQ_3:25;
then
A32: lb > 0 by A17,A28,FINSEQ_3:25;
then
A33: (l-'1)+1=(n-1)*lb+(m-1)+1 by NAT_D:2;
A34: k <= la*lb by A17,A28,FINSEQ_3:25;
then
A35: lb divides la*lb & 1 <= la*lb by A31,NAT_1:14,NAT_D:def 3;
1 <= lb by A32,NAT_1:14;
then
A36: (la*lb -' 1) div lb = la*lb div lb - 1 by A35,NAT_2:15;
set j=(k-'1) mod lb + 1;
set i=(k-'1) div lb + 1;
FG.k = F.i /\ G.j by A18,A28;
then
A37: FG.k /\ FG.l= (F.i /\ G.j) /\ (F.n /\ G.m) by A18,A29
.= F.i /\ (G.j /\ (F.n /\ G.m)) by XBOOLE_1:16
.= F.i /\ (F.n /\ (G.j /\ G.m)) by XBOOLE_1:16
.= (F.i /\ F.n) /\ (G.j /\ G.m) by XBOOLE_1:16;
l <= la*lb by A17,A29,FINSEQ_3:25;
then l -' 1 <= la*lb -' 1 by NAT_D:42;
then (l -' 1) div lb <= la*lb div lb - 1 by A36,NAT_2:24;
then (l -' 1) div lb + 1 <= la*lb div lb by XREAL_1:19;
then n >= 1 & n <= la by A32,NAT_1:14,NAT_D:18;
then n in Seg la;
then
A38: n in dom F by FINSEQ_1:def 3;
1 <= l by A29,FINSEQ_3:25;
then
A39: l - 1 + 1 = (n-1)*lb+m by A33,XREAL_1:233;
(l -' 1) mod lb < lb by A32,NAT_D:1;
then m >= 1 & m <= lb by NAT_1:13,14;
then m in Seg lb;
then
A40: m in dom G by FINSEQ_1:def 3;
k -' 1 <= la*lb -' 1 by A34,NAT_D:42;
then (k -' 1) div lb <= la*lb div lb - 1 by A36,NAT_2:24;
then (k -' 1) div lb + 1 <= la*lb div lb by XREAL_1:19;
then i >= 1 & i <= la by A32,NAT_1:11,NAT_D:18;
then i in Seg la;
then
A41: i in dom F by FINSEQ_1:def 3;
(k-'1)+1=(i-1)*lb+(j-1)+1 by A32,NAT_D:2;
then
A42: k - 1 + 1 = (i-1)*lb + j by A31,XREAL_1:233;
(k -' 1) mod lb < lb by A32,NAT_D:1;
then j >= 1 & j <= lb by NAT_1:11,13;
then j in Seg lb;
then
A43: j in dom G by FINSEQ_1:def 3;
per cases by A30,A42,A39;
suppose
i <> n;
then F.i misses F.n by A41,A38,MESFUNC3:4;
then FG.k /\ FG.l= {} /\ (G.j /\ G.m) by A37;
hence thesis;
end;
suppose
j <> m;
then G.j misses G.m by A43,A40,MESFUNC3:4;
then FG.k /\ FG.l= (F.i /\ F.n) /\ {} by A37;
hence thesis;
end;
end;
then reconsider FG as Finite_Sep_Sequence of S by MESFUNC3:4;
A44: dom f = union rng G by A15,COMSEQ_3:def 4;
A45: dom f = union rng FG
proof
thus dom f c= union rng FG
proof
let z be object;
assume
A46: z in dom f;
then consider Y be set such that
A47: z in Y and
A48: Y in rng F by A14,TARSKI:def 4;
consider Z be set such that
A49: z in Z and
A50: Z in rng G by A44,A46,TARSKI:def 4;
consider j be Nat such that
A51: j in dom G and
A52: Z = G.j by A50,FINSEQ_2:10;
consider i be Nat such that
A53: i in dom F and
A54: F.i = Y by A48,FINSEQ_2:10;
1 <= i by A53,FINSEQ_3:25;
then consider i9 being Nat such that
A55: i = 1 + (i9 qua Complex) by NAT_1:10;
set k=(i-1)*lb+j;
reconsider k as Nat by A55;
i <= la by A53,FINSEQ_3:25;
then i-1 <= la-1 by XREAL_1:9;
then (i-1)*lb <= (la - 1)*lb by XREAL_1:64;
then
A56: k <= (la - 1) * lb + j by XREAL_1:6;
A57: j <= lb by A51,FINSEQ_3:25;
then (la - 1) * lb + j <= (la - 1) * lb + lb by XREAL_1:6;
then
A58: k <= la*lb by A56,XXREAL_0:2;
A59: 1 <= j by A51,FINSEQ_3:25;
then consider j9 being Nat such that
A60: j = 1 + (j9 qua Complex) by NAT_1:10;
A61: j9 < lb by A57,A60,NAT_1:13;
A62: k >= j by A55,NAT_1:11;
then
A63: k -' 1 = k - 1 by A59,XREAL_1:233,XXREAL_0:2
.= i9*lb + j9 by A55,A60;
then
A64: i = (k-'1) div lb +1 by A55,A61,NAT_D:def 1;
A65: k >= 1 by A59,A62,XXREAL_0:2;
then
A66: k in Seg (la*lb) by A58;
A67: j = (k-'1) mod lb +1 by A60,A63,A61,NAT_D:def 2;
k in dom FG by A17,A65,A58,FINSEQ_3:25;
then
A68: FG.k in rng FG by FUNCT_1:def 3;
z in F.i /\ G.j by A47,A54,A49,A52,XBOOLE_0:def 4;
then z in FG.k by A18,A19,A64,A67,A66;
hence thesis by A68,TARSKI:def 4;
end;
let z be object;
assume z in union rng FG;
then consider Y be set such that
A69: z in Y and
A70: Y in rng FG by TARSKI:def 4;
consider k be Nat such that
A71: k in dom FG and
A72: FG.k = Y by A70,FINSEQ_2:10;
A73: 1 <= k by A71,FINSEQ_3:25;
then
A74: lb > 0 by A17,A71,FINSEQ_3:25;
then
A75: 1 <= lb by NAT_1:14;
A76: k <= la*lb by A17,A71,FINSEQ_3:25;
then lb divides (la*lb) & 1 <= la*lb by A73,NAT_1:14,NAT_D:def 3;
then
A77: (la*lb -' 1) div lb = la*lb div lb - 1 by A75,NAT_2:15;
set j=(k-'1) mod lb +1;
set i=(k-'1) div lb +1;
k -' 1 <= la*lb -' 1 by A76,NAT_D:42;
then (k -' 1) div lb <= la*lb div lb - 1 by A77,NAT_2:24;
then
A78: i >= 1 & i <= la*lb div lb by NAT_1:14,XREAL_1:19;
la*lb div lb = la by A74,NAT_D:18;
then i in dom F by A78,FINSEQ_3:25;
then
A79: F.i in rng F by FUNCT_1:def 3;
FG.k=F.i /\ G.j by A18,A71;
then z in F.i by A69,A72,XBOOLE_0:def 4;
hence thesis by A14,A79,TARSKI:def 4;
end;
for k being Nat,x,y being Element of X st k in dom FG & x in FG.k & y
in FG.k holds f.x = f.y
proof
let k be Nat;
let x,y be Element of X;
set i=(k-'1) div lb + 1;
set j=(k-'1) mod lb + 1;
assume that
A80: k in dom FG and
A81: x in FG.k & y in FG.k;
A82: FG.k c= union rng FG by A80,MESFUNC3:7;
then FG.k c= dom Im f by A45,COMSEQ_3:def 4;
then
A83: (Im f).x = Im(f.x) & (Im f).y = Im(f.y) by A81,COMSEQ_3:def 4;
A84: 1 <= k by A80,FINSEQ_3:25;
then
A85: lb > 0 by A17,A80,FINSEQ_3:25;
then (k -' 1) mod lb < lb by NAT_D:1;
then j >= 1 & j <= lb by NAT_1:13,14;
then
A86: j in dom G by FINSEQ_3:25;
FG.k c= dom Re f by A45,A82,COMSEQ_3:def 3;
then
A87: (Re f).x = Re(f.x) & (Re f).y = Re(f.y) by A81,COMSEQ_3:def 3;
A88: k <= la*lb by A17,A80,FINSEQ_3:25;
then
A89: k -' 1 <= la*lb -' 1 by NAT_D:42;
A90: FG.k = F.( (k-'1) div lb + 1 ) /\ G.( (k-'1) mod lb + 1 ) by A18,A80;
then x in G.j & y in G.j by A81,XBOOLE_0:def 4;
then
A91: Im(f.x) = Im(f.y) by A16,A86,A83;
A92: lb divides (la*lb) & 1 <= la*lb by A84,A88,NAT_1:14,NAT_D:def 3;
1 <= lb by A85,NAT_1:14;
then ((la*lb) -' 1) div lb = la*lb div lb - 1 by A92,NAT_2:15;
then (k -' 1) div lb <= la*lb div lb - 1 by A89,NAT_2:24;
then i <= la*lb div lb by XREAL_1:19;
then i >= 1 & i <= la by A85,NAT_1:14,NAT_D:18;
then
A93: i in dom F by FINSEQ_3:25;
x in F.i & y in F.i by A81,A90,XBOOLE_0:def 4;
then Re(f.x) = Re(f.y) by A13,A93,A87;
hence thesis by A91;
end;
hence thesis by A45;
end;
theorem Th44:
f is_simple_func_in S implies ex F be Finite_Sep_Sequence of S,
a be FinSequence of COMPLEX st dom f = union rng F & dom F= dom a & for n be
Nat st n in dom F for x be set st x in F.n holds f.x = a.n
proof
assume f is_simple_func_in S;
then consider F be Finite_Sep_Sequence of S such that
A1: dom f = union rng F and
A2: for n being Nat,x,y being Element of X st n in dom F & x in F.n & y
in F.n holds f.x = f.y;
defpred P[set ,set] means for x be set st x in F.$1 holds $2 = f.x;
A3: for k be Nat st k in Seg len F ex y be Element of COMPLEX st P[k,y]
proof
let k be Nat;
assume k in Seg len F;
then
A4: k in dom F by FINSEQ_1:def 3;
then
A5: F.k in rng F by FUNCT_1:3;
per cases;
suppose
A6: F.k = {};
0 in REAL by XREAL_0:def 1;
then reconsider y = 0 as Element of COMPLEX by NUMBERS:11;
take y;
thus thesis by A6;
end;
suppose
F.k <> {};
then consider x1 be object such that
A7: x1 in F.k by XBOOLE_0:def 1;
x1 in dom f by A1,A5,A7,TARSKI:def 4;
then f.x1 in rng f by FUNCT_1:3;
then reconsider y = f.x1 as Element of COMPLEX;
take y;
hereby
let x be set;
A8: rng F c= bool X by XBOOLE_1:1;
assume x in F.k;
hence y = f.x by A2,A4,A5,A7,A8;
end;
end;
end;
consider a be FinSequence of COMPLEX such that
A9: dom a = Seg len F & for k be Nat st k in Seg len F holds P[k,a.k]
from FINSEQ_1:sch 5(A3);
take F,a;
now
let n be Nat;
assume n in dom F;
then n in Seg len F by FINSEQ_1:def 3;
hence for x be set st x in F.n holds a.n = f.x by A9;
end;
hence thesis by A1,A9,FINSEQ_1:def 3;
end;
theorem Th45:
f is_simple_func_in S iff ex F be Finite_Sep_Sequence of S, a be
FinSequence of COMPLEX st F,a are_Re-presentation_of f
proof
hereby
assume f is_simple_func_in S;
then consider
F being Finite_Sep_Sequence of S, a be FinSequence of COMPLEX
such that
A1: dom f = union rng F & dom F= dom a & for n be Nat st n in dom F
for x be set st x in F.n holds f.x=a.n by Th44;
take F,a;
thus F,a are_Re-presentation_of f by A1;
end;
given F being Finite_Sep_Sequence of S, a be FinSequence of COMPLEX such
that
A2: F,a are_Re-presentation_of f;
A3: for n being Nat,x,y being Element of X st n in dom F & x in F.n & y in F
.n holds f.x = f.y
proof
let n being Nat,x,y being Element of X;
assume that
A4: n in dom F and
A5: x in F.n and
A6: y in F.n;
f.x=a.n by A2,A4,A5;
hence thesis by A2,A4,A6;
end;
dom f = union rng F by A2;
hence thesis by A3;
end;
reserve c for FinSequence of COMPLEX;
theorem Th46:
for n be Nat st n in dom Re c holds (Re c).n = Re(c.n)
proof
let n be Nat;
A1: (1/2*c*').n = 1/2*(c*'.n) by COMPLSP2:16;
len(1/2*c) = len c & len(1/2*(c*')) = len(c*') by COMPLSP2:3;
then
A2: len(1/2*c) = len(1/2*(c*')) by COMPLSP2:def 1;
len(c*') = len c by COMPLSP2:def 1;
then n in NAT & 1/2*(c + c*') = 1/2*c + 1/2*c*' by COMPLSP2:30
,ORDINAL1:def 12;
then
A3: (Re c).n = (1/2*c).n + (1/2*c*').n by A2,COMPLSP2:26;
assume
A4: n in dom Re c;
then n <= len Re c by FINSEQ_3:25;
then
A5: n <= len c by COMPLSP2:48;
1 <= n by A4,FINSEQ_3:25;
then (1/2*c*').n = 1/2*(c.n)*' by A5,A1,COMPLSP2:def 1;
then
A6: (Re c).n = 1/2*(c.n) + 1/2*(c.n)*' by A3,COMPLSP2:16;
c.n = Re(c.n) + (Im(c.n))** by COMPLEX1:13;
hence thesis by A6;
end;
theorem Th47:
for n be Nat st n in dom Im c holds (Im c).n = Im(c.n)
proof
let n be Nat;
assume
A1: n in dom Im c;
then
A2: 1 <= n by FINSEQ_3:25;
n <= len Im c by A1,FINSEQ_3:25;
then
A3: n <= len c by COMPLSP2:48;
A4: ((-1/2***)*c*').n = (-1/2***)*(c*'.n) by COMPLSP2:16
.= (-1/2***)*(c.n)*' by A2,A3,COMPLSP2:def 1;
len( (-1/2***)*c ) = len c & len( (-1/2***)*(c*') ) = len(c*') by
COMPLSP2:3;
then
A5: len( (-1/2***)*c ) = len( (-1/2***)*(c*') ) by COMPLSP2:def 1;
len(c*') = len c by COMPLSP2:def 1;
then n in NAT & (-1/2***)*(c - c*') = (-1/2***)*c - (-1/2***)*c*' by
COMPLSP2:43,ORDINAL1:def 12;
then (Im c).n = ((-1/2***)*c).n - ((-1/2***)*c*').n by A5,COMPLSP2:25;
then
A6: (Im c).n = (-1/2***)*(c.n) - (-1/2***)*(c.n)*' by A4,COMPLSP2:16;
c.n - (c.n)*' = Re(c.n) + (Im(c.n))*** -(Re(c.n) - (Im(c.n))***) by
COMPLEX1:13;
hence thesis by A6;
end;
theorem Th48:
for F be Finite_Sep_Sequence of S, a be FinSequence of COMPLEX
holds F,a are_Re-presentation_of f iff F,Re a are_Re-presentation_of Re f & F,
Im a are_Re-presentation_of Im f
proof
let F be Finite_Sep_Sequence of S, a be FinSequence of COMPLEX;
hereby
assume
A1: F,a are_Re-presentation_of f;
len Im a = len a by COMPLSP2:48;
then dom Im a = Seg len a by FINSEQ_1:def 3;
then dom Im a = dom a by FINSEQ_1:def 3;
then
A2: dom F = dom Im a by A1;
dom Im f = dom f by COMSEQ_3:def 4;
then
A3: dom Im f = union rng F by A1;
A4: for n be Nat st n in dom F for x be set st x in F.n holds (Im f).x = Im a.n
proof
let n be Nat;
assume
A5: n in dom F;
let x be set;
assume
A6: x in F.n;
F.n c= union rng F by A5,MESFUNC3:7;
then x in dom Im f by A3,A6;
then
A7: (Im f).x = Im(f.x) by COMSEQ_3:def 4;
Im(f.x) = Im(a.n) by A1,A5,A6;
hence thesis by A2,A5,A7,Th47;
end;
len Re a = len a by COMPLSP2:48;
then dom Re a = Seg len a by FINSEQ_1:def 3;
then dom Re a = dom a by FINSEQ_1:def 3;
then
A8: dom F = dom Re a by A1;
dom Re f = dom f by COMSEQ_3:def 3;
then
A9: dom Re f = union rng F by A1;
for n be Nat st n in dom F for x be set st x in F.n holds (Re f).x = Re a.n
proof
let n be Nat;
assume
A10: n in dom F;
let x be set;
assume
A11: x in F.n;
F.n c= union rng F by A10,MESFUNC3:7;
then x in dom Re f by A9,A11;
then
A12: (Re f).x = Re(f.x) by COMSEQ_3:def 3;
Re(f.x) = Re(a.n) by A1,A10,A11;
hence thesis by A8,A10,A12,Th46;
end;
hence
F,Re a are_Re-presentation_of Re f & F,Im a are_Re-presentation_of Im
f by A9,A3,A8,A2,A4;
end;
assume that
A13: F,Re a are_Re-presentation_of Re f and
A14: F,Im a are_Re-presentation_of Im f;
A15: dom F = dom Re a by A13;
A16: dom Re f = union rng F by A13;
then
A17: dom f = union rng F by COMSEQ_3:def 3;
A18: dom F = dom Im a by A14;
A19: dom Im f = union rng F by A14;
A20: for n be Nat st n in dom F for x be set st x in F.n holds f.x = a.n
proof
let n be Nat;
assume
A21: n in dom F;
let x be set;
assume
A22: x in F.n;
A23: F.n c= union rng F by A21,MESFUNC3:7;
then x in dom Im f by A19,A22;
then
A24: (Im f).x = Im(f.x) by COMSEQ_3:def 4;
x in dom Re f by A16,A22,A23;
then
A25: (Re f).x = Re(f.x) by COMSEQ_3:def 3;
(Im f).x = Im a.n by A14,A21,A22;
then
A26: Im(f.x) = Im(a.n) by A18,A21,A24,Th47;
(Re f).x = Re a.n by A13,A21,A22;
then Re(f.x) = Re(a.n) by A15,A21,A25,Th46;
hence thesis by A26;
end;
len Re a = len a by COMPLSP2:48;
then dom Re a = Seg len a by FINSEQ_1:def 3;
then dom F = dom a by A15,FINSEQ_1:def 3;
hence thesis by A17,A20;
end;
theorem
f is_simple_func_in S iff ex F be Finite_Sep_Sequence of S, c be
FinSequence of COMPLEX st dom f = union rng F & dom F = dom c & (for n be Nat
st n in dom F for x be set st x in F.n holds (Re f).x = Re c.n) & for n be Nat
st n in dom F for x be set st x in F.n holds (Im f).x = Im c.n
proof
hereby
assume f is_simple_func_in S;
then consider
F be Finite_Sep_Sequence of S, c be FinSequence of COMPLEX such
that
A1: F,c are_Re-presentation_of f by Th45;
F,Im c are_Re-presentation_of Im f by A1,Th48;
then
A2: for n be Nat st n in dom F for x be set st x in F.n holds (Im f).x =
Im c.n;
F,Re c are_Re-presentation_of Re f by A1,Th48;
then
A3: for n be Nat st n in dom F for x be set st x in F.n holds (Re f).x =
Re c.n;
dom f = union rng F & dom F = dom c by A1;
hence ex F be Finite_Sep_Sequence of S, c be FinSequence of COMPLEX st dom
f = union rng F & dom F = dom c & (for n be Nat st n in dom F for x be set st x
in F.n holds (Re f).x = Re c.n) & for n be Nat st n in dom F for x be set st x
in F.n holds (Im f).x = Im c.n by A3,A2;
end;
given F be Finite_Sep_Sequence of S, c be FinSequence of COMPLEX such that
A4: dom f = union rng F and
A5: dom F = dom c and
A6: for n be Nat st n in dom F for x be set st x in F.n holds (Re f).x
= Re c.n and
A7: for n be Nat st n in dom F for x be set st x in F.n holds (Im f).x
= Im c.n;
A8: dom Im f = union rng F by A4,COMSEQ_3:def 4;
len Im c = len c by COMPLSP2:48;
then dom Im c = Seg len c by FINSEQ_1:def 3;
then
A9: dom F = dom Im c by A5,FINSEQ_1:def 3;
len Re c = len c by COMPLSP2:48;
then dom Re c = Seg len c by FINSEQ_1:def 3;
then
A10: dom F = dom Re c by A5,FINSEQ_1:def 3;
A11: dom Re f = union rng F by A4,COMSEQ_3:def 3;
for n be Nat st n in dom F for x be set st x in F.n holds f.x = c.n
proof
let n be Nat;
assume
A12: n in dom F;
let x be set;
assume
A13: x in F.n;
A14: F.n c= union rng F by A12,MESFUNC3:7;
then x in dom Im f by A8,A13;
then
A15: (Im f).x = Im(f.x) by COMSEQ_3:def 4;
x in dom Re f by A11,A13,A14;
then
A16: (Re f).x = Re(f.x) by COMSEQ_3:def 3;
(Im f).x = Im c.n by A7,A12,A13;
then
A17: Im(f.x) = Im(c.n) by A9,A12,A15,Th47;
(Re f).x = Re c.n by A6,A12,A13;
then Re(f.x) = Re(c.n) by A10,A12,A16,Th46;
hence thesis by A17;
end;
then F,c are_Re-presentation_of f by A4,A5;
hence thesis by Th45;
end;
*