MESFUN7C: The Measurability of Complex-Valued Functional Sequences

:: The Measurability of Complex-Valued Functional Sequences
:: by Keiko Narita , Noboru Endou and Yasunari Shidama
::
:: Received December 16, 2008
:: Copyright (c) 2008-2021 Association of Mizar Users

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 :: MESFUN7C:def 1
f;

proof end;

end;

:: deftheorem defines R_EAL MESFUN7C:def 1 :

theorem Th1: :: MESFUN7C:1

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X holds f # x = (R_EAL f) # x

proof end;

registration

let X be non empty set ;
let f be Function of X,REAL;

cluster R_EAL f -> total ;

end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func inf f -> PartFunc of X,ExtREAL equals :: MESFUN7C:def 2
inf (R_EAL f);

end;

:: deftheorem defines inf MESFUN7C:def 2 :

theorem Th2: :: MESFUN7C:2

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (inf f) holds
(inf f) . x = inf (rng (R_EAL (f # x)))

proof end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func sup f -> PartFunc of X,ExtREAL equals :: MESFUN7C:def 3
sup (R_EAL f);

end;

:: deftheorem defines sup MESFUN7C:def 3 :

theorem Th3: :: MESFUN7C:3

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (sup f) holds
(sup f) . x = sup (rng (R_EAL (f # x)))

proof end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func inferior_realsequence f -> with_the_same_dom Functional_Sequence of X,ExtREAL equals :: MESFUN7C:def 4
inferior_realsequence (R_EAL f);

end;

:: deftheorem defines inferior_realsequence MESFUN7C:def 4 :

theorem Th4: :: MESFUN7C:4

for X being non empty set
for f being Functional_Sequence of X,REAL
for n being Nat holds
( dom ((inferior_realsequence f) . n) = dom (f . 0) & ( for x being 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 end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func superior_realsequence f -> with_the_same_dom Functional_Sequence of X,ExtREAL equals :: MESFUN7C:def 5
superior_realsequence (R_EAL f);

end;

:: deftheorem defines superior_realsequence MESFUN7C:def 5 :

theorem Th5: :: MESFUN7C:5

for X being non empty set
for f being Functional_Sequence of X,REAL
for n being Nat holds
( dom ((superior_realsequence f) . n) = dom (f . 0) & ( for x being 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 end;

theorem :: MESFUN7C:6

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (f . 0) holds
(inferior_realsequence f) # x = inferior_realsequence (R_EAL (f # x))

proof end;

registration

let X be non empty set ;
let f be with_the_same_dom Functional_Sequence of X,REAL;

cluster R_EAL f -> with_the_same_dom ;

proof end;

end;

theorem Th7: :: MESFUN7C:7

for X being non empty set
for f being with_the_same_dom Functional_Sequence of X,REAL
for S being SigmaField of X
for E being Element of S
for n being Nat st f . n is E -measurable holds
(R_EAL f) . n is E -measurable

proof end;

theorem :: MESFUN7C:8

for X being non empty set
for f being Functional_Sequence of X,REAL
for n being Nat holds (R_EAL f) ^\ n = R_EAL (f ^\ n) ;

theorem :: MESFUN7C:9

for X being non empty set
for f being with_the_same_dom Functional_Sequence of X,REAL
for n being Nat holds (inferior_realsequence f) . n = inf (f ^\ n) by MESFUNC8:8;

theorem :: MESFUN7C:10

for X being non empty set
for f being with_the_same_dom Functional_Sequence of X,REAL
for n being Nat holds (superior_realsequence f) . n = sup (f ^\ n) by MESFUNC8:9;

theorem Th11: :: MESFUN7C:11

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (f . 0) holds
(superior_realsequence f) # x = superior_realsequence (R_EAL (f # x))

proof end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func lim_inf f -> PartFunc of X,ExtREAL equals :: MESFUN7C:def 6
lim_inf (R_EAL f);

end;

:: deftheorem defines lim_inf MESFUN7C:def 6 :

theorem Th12: :: MESFUN7C:12

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (lim_inf f) holds
(lim_inf f) . x = lim_inf (R_EAL (f # x))

proof end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func lim_sup f -> PartFunc of X,ExtREAL equals :: MESFUN7C:def 7
lim_sup (R_EAL f);

end;

:: deftheorem defines lim_sup MESFUN7C:def 7 :

theorem Th13: :: MESFUN7C:13

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (lim_sup f) holds
(lim_sup f) . x = lim_sup (R_EAL (f # x))

proof end;

definition

let X be non empty set ;
let f be Functional_Sequence of X,REAL;

func lim f -> PartFunc of X,ExtREAL equals :: MESFUN7C:def 8
lim (R_EAL f);

end;

:: deftheorem defines lim MESFUN7C:def 8 :

theorem Th14: :: MESFUN7C:14

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (lim f) holds
(lim f) . x = lim (R_EAL (f # x))

proof end;

theorem Th15: :: MESFUN7C:15

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being 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 end;

theorem :: MESFUN7C:16

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for F being SetSequence of S
for r being Real st ( for n being 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 end;

theorem :: MESFUN7C:17

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for F being SetSequence of S
for r being Real st ( for n being 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 end;

theorem Th18: :: MESFUN7C:18

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) holds
lim_sup f is E -measurable

proof end;

theorem :: MESFUN7C:19

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) holds
lim_inf f is E -measurable

proof end;

theorem :: MESFUN7C:20

for X being non empty set
for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (f . 0) & f # x is convergent holds
(superior_realsequence f) # x is bounded_below

proof end;

theorem Th21: :: MESFUN7C:21

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is E -measurable

proof end;

theorem Th22: :: MESFUN7C:22

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for g being PartFunc of X,ExtREAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) holds
g is E -measurable

proof end;

definition

let X be non empty set ;
let H be Functional_Sequence of X,COMPLEX;
let x be Element of X;

func H # x -> Complex_Sequence means :Def9: :: MESFUN7C:def 9
for n being Nat holds it . n = (H . n) . x;

proof end;

proof end;

end;

:: deftheorem Def9 defines # MESFUN7C:def 9 :

definition

let X be non empty set ;
let f be Functional_Sequence of X,COMPLEX;

func lim f -> PartFunc of X,COMPLEX means :Def10: :: MESFUN7C:def 10
( dom it = dom (f . 0) & ( for x being Element of X st x in dom it holds
it . x = lim (f # x) ) );

proof end;

proof end;

end;

:: deftheorem Def10 defines lim MESFUN7C:def 10 :

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: :: MESFUN7C:def 11
for n being Nat holds
( dom (it . n) = dom (f . n) & ( for x being Element of X st x in dom (it . n) holds
(it . n) . x = (Re (f # x)) . n ) );

proof end;

proof end;

end;

:: deftheorem Def11 defines Re MESFUN7C:def 11 :

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 ;

proof 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: :: MESFUN7C:def 12
for n being Nat holds
( dom (it . n) = dom (f . n) & ( for x being Element of X st x in dom (it . n) holds
(it . n) . x = (Im (f # x)) . n ) );

proof end;

proof end;

end;

:: deftheorem Def12 defines Im MESFUN7C:def 12 :

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 ;

proof end;

end;

theorem Th23: :: MESFUN7C:23

for X being non empty set
for f being with_the_same_dom Functional_Sequence of X,COMPLEX
for x being Element of X st x in dom (f . 0) holds
( (Re f) # x = Re (f # x) & (Im f) # x = Im (f # x) )

proof end;

theorem Th24: :: MESFUN7C:24

for X being non empty set
for f being Functional_Sequence of X,COMPLEX
for n being Nat holds
( (Re f) . n = Re (f . n) & (Im f) . n = Im (f . n) )

proof end;

theorem Th25: :: MESFUN7C:25

for X being non empty set
for f being with_the_same_dom Functional_Sequence of X,COMPLEX st ( for x being 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 end;

theorem :: MESFUN7C:26

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,COMPLEX
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is E -measurable

proof end;

theorem :: MESFUN7C:27

for X being non empty set
for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,COMPLEX
for g being PartFunc of X,COMPLEX
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) holds
g is E -measurable

proof end;

theorem :: MESFUN7C:28

for X being non empty set
for Y being set
for f being PartFunc of X,COMPLEX
for r being Real holds (r (#) f) | Y = r (#) (f | Y)

proof end;

Lm1: for X being non empty set
for f being PartFunc of X,COMPLEX holds |.f.| is nonnegative

proof end;

theorem :: MESFUN7C:29

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for k being Real
for E being Element of S st 0 <= k & E c= dom f & f is E -measurable holds
|.f.| to_power k is E -measurable

proof end;

theorem Th30: :: MESFUN7C:30

for X being non empty set
for f, g being PartFunc of X,REAL holds (R_EAL f) (#) (R_EAL g) = R_EAL (f (#) g)

proof end;

theorem Th31: :: MESFUN7C:31

for X being non empty set
for S being SigmaField of X
for E being Element of S
for f, g being PartFunc of X,REAL st (dom f) /\ (dom g) = E & f is E -measurable & g is E -measurable holds
f (#) g is E -measurable

proof end;

theorem Th32: :: MESFUN7C:32

for X being non empty set
for f, g being PartFunc of X,COMPLEX holds
( Re (f (#) g) = ((Re f) (#) (Re g)) - ((Im f) (#) (Im g)) & Im (f (#) g) = ((Im f) (#) (Re g)) + ((Re f) (#) (Im g)) )

proof end;

theorem :: MESFUN7C:33

for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX
for E being Element of S st (dom f) /\ (dom g) = E & f is E -measurable & g is E -measurable holds
f (#) g is E -measurable

proof end;

theorem Th34: :: MESFUN7C:34

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st ex E being Element of S st
( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds
g . x <= f . x ) holds
Integral (M,g) <= Integral (M,f)

proof end;

theorem Th35: :: MESFUN7C:35

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,COMPLEX st f is_integrable_on M holds
( ex A being Element of S st
( A = dom f & f is A -measurable ) & |.f.| is_integrable_on M )

proof end;

theorem :: MESFUN7C:36

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,COMPLEX st f is_integrable_on M holds
ex F being sequence of S st
( ( for n being Nat holds F . n = (dom f) /\ (great_eq_dom (|.f.|,(1 / (n + 1)))) ) & (dom f) \ (eq_dom (|.f.|,0)) = union (rng F) & ( for n being Nat holds
( F . n in S & M . (F . n) < +infty ) ) )

proof end;

theorem Th37: :: MESFUN7C:37

for X being non empty set
for f being PartFunc of X,COMPLEX
for A being set holds |.f.| | A = |.(f | A).|

proof end;

theorem Th38: :: MESFUN7C:38

for X being non empty set
for f, g being PartFunc of X,COMPLEX holds
( dom (|.f.| + |.g.|) = (dom f) /\ (dom g) & dom |.(f + g).| c= dom |.f.| )

proof end;

theorem Th39: :: MESFUN7C:39

for X being non empty set
for f, g being PartFunc of X,COMPLEX holds (|.f.| | (dom |.(f + g).|)) + (|.g.| | (dom |.(f + g).|)) = (|.f.| + |.g.|) | (dom |.(f + g).|)

proof end;

theorem Th40: :: MESFUN7C:40

for X being non empty set
for f, g being PartFunc of X,COMPLEX
for x being set st x in dom |.(f + g).| holds
|.(f + g).| . x <= (|.f.| + |.g.|) . x

proof end;

theorem Th41: :: MESFUN7C:41

for X being non empty set
for f, g being PartFunc of X,REAL st ( for x being set st x in dom f holds
f . x <= g . x ) holds
g - f is nonnegative

proof end;

theorem :: MESFUN7C:42

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,COMPLEX st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = dom (f + g) & Integral (M,(|.(f + g).| | E)) <= (Integral (M,(|.f.| | E))) + (Integral (M,(|.g.| | E))) )

proof end;

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 :: MESFUN7C:def 13
ex F being Finite_Sep_Sequence of S st
( dom f = union (rng F) & ( for n being Nat
for 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;

:: deftheorem defines is_simple_func_in MESFUN7C:def 13 :

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 :: MESFUN7C:def 14
( dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
f . x = a . n ) );

end;

:: deftheorem defines are_Re-presentation_of MESFUN7C:def 14 :

definition

let X be non empty set ;
let S be SigmaField of X;
let f be PartFunc of X,COMPLEX;
let F be Finite_Sep_Sequence of S;
let a be FinSequence of COMPLEX ;

pred F,a are_Re-presentation_of f means :: MESFUN7C:def 15
( dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
f . x = a . n ) );

end;

:: deftheorem defines are_Re-presentation_of MESFUN7C:def 15 :

theorem :: MESFUN7C:43

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX holds
( f is_simple_func_in S iff ( Re f is_simple_func_in S & Im f is_simple_func_in S ) )

proof end;

theorem Th44: :: MESFUN7C:44

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX st f is_simple_func_in S holds
ex F being Finite_Sep_Sequence of S ex a being FinSequence of COMPLEX st
( dom f = union (rng F) & dom F = dom a & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
f . x = a . n ) )

proof end;

theorem Th45: :: MESFUN7C:45

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX holds
( f is_simple_func_in S iff ex F being Finite_Sep_Sequence of S ex a being FinSequence of COMPLEX st F,a are_Re-presentation_of f )

proof end;

theorem Th46: :: MESFUN7C:46

for c being FinSequence of COMPLEX
for n being Nat st n in dom (Re c) holds
(Re c) . n = Re (c . n)

proof end;

theorem Th47: :: MESFUN7C:47

for c being FinSequence of COMPLEX
for n being Nat st n in dom (Im c) holds
(Im c) . n = Im (c . n)

proof end;

theorem Th48: :: MESFUN7C:48

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for F being Finite_Sep_Sequence of S
for a being 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 end;

theorem :: MESFUN7C:49

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX holds
( f is_simple_func_in S iff ex F being Finite_Sep_Sequence of S ex c being FinSequence of COMPLEX st
( dom f = union (rng F) & dom F = dom c & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
(Re f) . x = (Re c) . n ) & ( for n being Nat st n in dom F holds
for x being set st x in F . n holds
(Im f) . x = (Im c) . n ) ) )

proof end;