:: Integral of Complex-Valued Measurable Function
:: by Keiko Narita , Noboru Endou and Yasunari Shidama
::
:: Received July 30, 2008
:: Copyright (c) 2008-2021 Association of Mizar Users

theorem :: MESFUN6C:1

for a, b being Real holds
( a + b = a + b & - a = - a & a - b = a - b & a * b = a * b ) ;

definition

let X be non empty set ;
let S be SigmaField of X;
let E be Element of S;
let f be PartFunc of X,COMPLEX;

attr f is E -measurable means :: MESFUN6C:def 1
( Re f is E -measurable & Im f is E -measurable );

end;

:: deftheorem defines -measurable MESFUN6C:def 1 :
for X being non empty set
for S being SigmaField of X
for E being Element of S
for f being PartFunc of X,COMPLEX holds
( f is E -measurable iff ( Re f is E -measurable & Im f is E -measurable ) );

theorem Th2: :: MESFUN6C:2

for X being non empty set
for f being PartFunc of X,COMPLEX
for r being Real holds
( r (#) (Re f) = Re (r (#) f) & r (#) (Im f) = Im (r (#) f) )

proof end;

theorem Th3: :: MESFUN6C:3

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

proof end;

theorem Th4: :: MESFUN6C:4

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

proof end;

theorem Th5: :: MESFUN6C:5

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

proof end;

theorem Th6: :: MESFUN6C:6

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

proof end;

theorem Th7: :: MESFUN6C:7

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A being Element of S holds
( (Re f) | A = Re (f | A) & (Im f) | A = Im (f | A) )

proof end;

theorem Th8: :: MESFUN6C:8

for X being non empty set
for f being PartFunc of X,COMPLEX holds f = (Re f) + (<i> (#) (Im f))

proof end;

theorem :: MESFUN6C:9

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A, B being Element of S st B c= A & f is A -measurable holds
f is B -measurable by MESFUNC6:16;

theorem :: MESFUN6C:10

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A, B being Element of S st f is A -measurable & f is B -measurable holds
f is A \/ B -measurable by MESFUNC6:17;

theorem :: MESFUN6C:11

for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX
for A being Element of S st f is A -measurable & g is A -measurable holds
f + g is A -measurable

proof end;

theorem :: MESFUN6C:12

for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX
for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable

proof end;

theorem :: MESFUN6C:13

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

proof end;

theorem :: MESFUN6C:14

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

proof end;

theorem :: MESFUN6C:15

for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX st dom f in S & dom g in S holds
dom (f + g) in S

proof end;

theorem :: MESFUN6C:16

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A, B being Element of S st dom f = A holds
( f is B -measurable iff f is A /\ B -measurable )

proof end;

theorem Th17: :: MESFUN6C:17

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for c being Complex
for A being Element of S st f is A -measurable & A c= dom f holds
c (#) f is A -measurable

proof end;

theorem :: MESFUN6C:18

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX st ex A being Element of S st dom f = A holds
for c being Complex
for B being Element of S st f is B -measurable holds
c (#) f is B -measurable

proof end;

definition

let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let f be PartFunc of X,COMPLEX;

pred f is_integrable_on M means :: MESFUN6C:def 2
( Re f is_integrable_on M & Im f is_integrable_on M );

end;

:: deftheorem defines is_integrable_on MESFUN6C:def 2 :
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 holds
( f is_integrable_on M iff ( Re f is_integrable_on M & Im f is_integrable_on M ) );

definition

let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let f be PartFunc of X,COMPLEX;
assume A1: f is_integrable_on M ;

func Integral (M,f) -> Complex means :Def3: :: MESFUN6C:def 3
ex R, I being Real st
( R = Integral (M,(Re f)) & I = Integral (M,(Im f)) & it = R + (I * <i>) );

existence
ex b₁ being Complex ex R, I being Real st
( R = Integral (M,(Re f)) & I = Integral (M,(Im f)) & b₁ = R + (I * <i>) )

proof end;

uniqueness
for b₁, b₂ being Complex st ex R, I being Real st
( R = Integral (M,(Re f)) & I = Integral (M,(Im f)) & b₁ = R + (I * <i>) ) & ex R, I being Real st
( R = Integral (M,(Re f)) & I = Integral (M,(Im f)) & b₂ = R + (I * <i>) ) holds
b₁ = b₂ ;

end;

:: deftheorem Def3 defines Integral MESFUN6C:def 3 :
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
for b₅ being Complex holds
( b₅ = Integral (M,f) iff ex R, I being Real st
( R = Integral (M,(Re f)) & I = Integral (M,(Im f)) & b₅ = R + (I * <i>) ) );

Lm1: 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,ExtREAL holds
( max+ f is nonnegative & max- f is nonnegative )

proof end;

theorem Th19: :: MESFUN6C:19

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,ExtREAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is E -measurable ) & M . A = 0 holds
f | A is_integrable_on M

proof end;

theorem Th20: :: MESFUN6C:20

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,REAL
for E, A being Element of S st ex E being Element of S st
( E = dom f & f is E -measurable ) & M . A = 0 holds
f | A is_integrable_on M by Th19;

theorem :: MESFUN6C:21

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
for A being Element of S st ex E being Element of S st
( E = dom f & f is E -measurable ) & M . A = 0 holds
( f | A is_integrable_on M & Integral (M,(f | A)) = 0 )

proof end;

theorem :: MESFUN6C:22

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
for E, A being Element of S st E = dom f & f is_integrable_on M & M . A = 0 holds
Integral (M,(f | (E \ A))) = Integral (M,f)

proof end;

theorem Th23: :: MESFUN6C:23

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
for A being Element of S st f is_integrable_on M holds
f | A is_integrable_on M

proof end;

theorem :: MESFUN6C:24

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
for A, B being Element of S st f is_integrable_on M & A misses B holds
Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))

proof end;

theorem :: MESFUN6C:25

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
for A, B being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )

proof end;

definition

let k be Real;
let X be non empty set ;
let f be PartFunc of X,REAL;

func f to_power k -> PartFunc of X,REAL means :Def4: :: MESFUN6C:def 4
( dom it = dom f & ( for x being Element of X st x in dom it holds
it . x = (f . x) to_power k ) );

existence
ex b₁ being PartFunc of X,REAL st
( dom b₁ = dom f & ( for x being Element of X st x in dom b₁ holds
b₁ . x = (f . x) to_power k ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of X,REAL st dom b₁ = dom f & ( for x being Element of X st x in dom b₁ holds
b₁ . x = (f . x) to_power k ) & dom b₂ = dom f & ( for x being Element of X st x in dom b₂ holds
b₂ . x = (f . x) to_power k ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines to_power MESFUN6C:def 4 :
for k being Real
for X being non empty set
for f, b₄ being PartFunc of X,REAL holds
( b₄ = f to_power k iff ( dom b₄ = dom f & ( for x being Element of X st x in dom b₄ holds
b₄ . x = (f . x) to_power k ) ) );

registration

let X be non empty set ;

cluster Relation-like X -defined REAL -valued Function-like V54() V55() V56() nonnegative for Element of K16(K17(X,REAL));

existence
ex b₁ being PartFunc of X,REAL st b₁ is nonnegative

proof end;

end;

registration

let k be non negative Real;
let X be non empty set ;
let f be nonnegative PartFunc of X,REAL;

cluster f to_power k -> nonnegative ;

coherence
f to_power k is nonnegative

proof end;

end;

theorem Th26: :: MESFUN6C:26

for k being Real
for X being non empty set
for S being SigmaField of X
for E being Element of S
for f being PartFunc of X,REAL st f is nonnegative & 0 <= k holds
f to_power k is nonnegative ;

theorem Th27: :: MESFUN6C:27

for x being object
for X being non empty set
for S being SigmaField of X
for E being Element of S
for f being PartFunc of X,REAL st f is nonnegative holds
(f . x) to_power (1 / 2) = sqrt (f . x)

proof end;

theorem Th28: :: MESFUN6C:28

for X being non empty set
for S being SigmaField of X
for A being Element of S
for f being PartFunc of X,REAL
for a being Real st A c= dom f holds
A /\ (less_dom (f,a)) = A \ (A /\ (great_eq_dom (f,a)))

proof end;

reconsider jj = 1 as Real ;

theorem Th29: :: MESFUN6C:29

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

proof end;

theorem Th30: :: MESFUN6C:30

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A being Element of S st f is A -measurable & A c= dom f holds
|.f.| is A -measurable

proof end;

theorem Th31: :: MESFUN6C:31

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 ex A being Element of S st
( A = dom f & f is A -measurable ) holds
( f is_integrable_on M iff |.f.| is_integrable_on M )

proof end;

theorem :: MESFUN6C:32

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
dom (f + g) in S

proof end;

theorem Th33: :: MESFUN6C:33

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
f + g is_integrable_on M

proof end;

theorem Th34: :: MESFUN6C: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 f is_integrable_on M & g is_integrable_on M holds
f - g is_integrable_on M

proof end;

theorem :: MESFUN6C:35

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
f - g is_integrable_on M

proof end;

theorem Th36: :: MESFUN6C:36

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) /\ (dom g) & Integral (M,(f + g)) = (Integral (M,(f | E))) + (Integral (M,(g | E))) )

proof end;

theorem :: MESFUN6C:37

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 f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral (M,(f - g)) = (Integral (M,(f | E))) + (Integral (M,((- g) | E))) )

proof end;

theorem Th38: :: MESFUN6C:38

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
for r being Real st f is_integrable_on M holds
( r (#) f is_integrable_on M & Integral (M,(r (#) f)) = r * (Integral (M,f)) )

proof end;

theorem Th39: :: MESFUN6C:39

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
( <i> (#) f is_integrable_on M & Integral (M,(<i> (#) f)) = <i> * (Integral (M,f)) )

proof end;

theorem Th40: :: MESFUN6C:40

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
for c being Complex st f is_integrable_on M holds
( c (#) f is_integrable_on M & Integral (M,(c (#) f)) = c * (Integral (M,f)) )

proof end;

Lm2: 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 ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M & Integral (M,f) = 0 holds
|.(Integral (M,f)).| <= Integral (M,|.f.|)

proof end;

theorem Th41: :: MESFUN6C:41

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

proof end;

theorem Th42: :: MESFUN6C: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,REAL st ex A being Element of S st
( A = (dom f) /\ (dom g) & f is A -measurable & g is A -measurable ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) )

proof end;

Lm3: 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 ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M & Integral (M,f) <> 0 holds
|.(Integral (M,f)).| <= Integral (M,|.f.|)

proof end;

theorem :: MESFUN6C:43

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 ex A being Element of S st
( A = dom f & f is A -measurable ) & f is_integrable_on M holds
|.(Integral (M,f)).| <= Integral (M,|.f.|)

proof end;

definition

let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let f be PartFunc of X,COMPLEX;
let B be Element of S;

func Integral_on (M,B,f) -> Complex equals :: MESFUN6C:def 5
Integral (M,(f | B));

coherence
Integral (M,(f | B)) is Complex ;

end;

:: deftheorem defines Integral_on MESFUN6C:def 5 :
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
for B being Element of S holds Integral_on (M,B,f) = Integral (M,(f | B));

theorem :: MESFUN6C:44

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
for B being Element of S st f is_integrable_on M & g is_integrable_on M & B c= dom (f + g) holds
( f + g is_integrable_on M & Integral_on (M,B,(f + g)) = (Integral_on (M,B,f)) + (Integral_on (M,B,g)) )

proof end;

theorem :: MESFUN6C:45

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
for c being Complex
for B being Element of S st f is_integrable_on M holds
Integral_on (M,B,(c (#) f)) = c * (Integral_on (M,B,f))

proof end;

theorem :: MESFUN6C:46

for X being non empty set
for S being SigmaField of X
for A being Element of S
for f being PartFunc of X,REAL
for a being Real st A c= dom f holds
A /\ (great_eq_dom (f,a)) = A \ (A /\ (less_dom (f,a)))

proof end;

theorem :: MESFUN6C:47

for X being non empty set
for S being SigmaField of X
for A being Element of S
for f being PartFunc of X,REAL
for a being Real st A c= dom f holds
A /\ (great_dom (f,a)) = A \ (A /\ (less_eq_dom (f,a)))

proof end;

theorem :: MESFUN6C:48

for X being non empty set
for S being SigmaField of X
for A being Element of S
for f being PartFunc of X,REAL
for a being Real st A c= dom f holds
A /\ (less_eq_dom (f,a)) = A \ (A /\ (great_dom (f,a)))

proof end;

theorem :: MESFUN6C:49

for X being non empty set
for S being SigmaField of X
for A being Element of S
for f being PartFunc of X,REAL
for a being Real holds A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))

proof end;