:: Lebesgue's Convergence Theorem of Complex-Valued Function
:: by Keiko Narita , Noboru Endou and Yasunari Shidama
::
:: Received March 17, 2009
:: Copyright (c) 2009-2011 Association of Mizar Users

begin

definition

let X, Y be set ;
let F be Functional_Sequence of X,Y;
let D be set ;
func F || D -> Functional_Sequence of X,Y means :Def1: :: MESFUN9C:def 1
for n being Nat holds it . n = (F . n) | D;
existence
ex b₁ being Functional_Sequence of X,Y st
for n being Nat holds b₁ . n = (F . n) | D

proof end;

uniqueness
for b₁, b₂ being Functional_Sequence of X,Y st ( for n being Nat holds b₁ . n = (F . n) | D ) & ( for n being Nat holds b₂ . n = (F . n) | D ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines || MESFUN9C:def 1 :
for X, Y being set
for F being Functional_Sequence of X,Y
for D being set
for b₅ being Functional_Sequence of X,Y holds
( b₅ = F || D iff for n being Nat holds b₅ . n = (F . n) | D );

theorem Th1: :: MESFUN9C:1

for X being non empty set
for F being Functional_Sequence of X,REAL
for x being Element of X
for D being set st x in D & F # x is convergent holds
(F || D) # x is convergent

proof end;

theorem Th2: :: MESFUN9C:2

for X, Y, D being set
for F being Functional_Sequence of X,Y st F is with_the_same_dom holds
F || D is with_the_same_dom

proof end;

theorem Th3: :: MESFUN9C:3

for X being non empty set
for F being Functional_Sequence of X,REAL
for D being set st D c= dom (F . 0) & ( for x being Element of X st x in D holds
F # x is convergent ) holds
(lim F) | D = lim (F || D)

proof end;

theorem Th4: :: MESFUN9C:4

for X being non empty set
for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,REAL
for n being Nat st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds F . m is_measurable_on E ) holds
(F || E) . n is_measurable_on E

proof end;

theorem Th5: :: MESFUN9C:5

for seq being Real_Sequence holds Partial_Sums (R_EAL seq) = R_EAL (Partial_Sums seq)

proof end;

theorem Th6: :: MESFUN9C:6

for X being non empty set
for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,REAL st ( for x being Element of X st x in E holds
F # x is summable ) holds
for x being Element of X st x in E holds
(F || E) # x is summable

proof end;

definition

let X be non empty set ;
let F be Functional_Sequence of X,REAL;
func Partial_Sums F -> Functional_Sequence of X,REAL means :Def2: :: MESFUN9C:def 2
( it . 0 = F . 0 & ( for n being Nat holds it . (n + 1) = (it . n) + (F . (n + 1)) ) );
existence
ex b₁ being Functional_Sequence of X,REAL st
( b₁ . 0 = F . 0 & ( for n being Nat holds b₁ . (n + 1) = (b₁ . n) + (F . (n + 1)) ) )

proof end;

uniqueness
for b₁, b₂ being Functional_Sequence of X,REAL st b₁ . 0 = F . 0 & ( for n being Nat holds b₁ . (n + 1) = (b₁ . n) + (F . (n + 1)) ) & b₂ . 0 = F . 0 & ( for n being Nat holds b₂ . (n + 1) = (b₂ . n) + (F . (n + 1)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines Partial_Sums MESFUN9C:def 2 :
for X being non empty set
for F, b₃ being Functional_Sequence of X,REAL holds
( b₃ = Partial_Sums F iff ( b₃ . 0 = F . 0 & ( for n being Nat holds b₃ . (n + 1) = (b₃ . n) + (F . (n + 1)) ) ) );

theorem Th7: :: MESFUN9C:7

for X being non empty set
for F being Functional_Sequence of X,REAL holds Partial_Sums (R_EAL F) = R_EAL (Partial_Sums F)

proof end;

theorem Th8: :: MESFUN9C:8

for X being non empty set
for F being Functional_Sequence of X,REAL
for n, m being Nat
for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds
( z in dom ((Partial_Sums F) . m) & z in dom (F . m) )

proof end;

theorem Th9: :: MESFUN9C:9

for X being non empty set
for F being Functional_Sequence of X,REAL holds R_EAL F is additive

proof end;

theorem Th10: :: MESFUN9C:10

for X being non empty set
for F being Functional_Sequence of X,REAL
for n being Nat holds dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n }

proof end;

theorem Th11: :: MESFUN9C:11

for X being non empty set
for F being Functional_Sequence of X,REAL
for n being Nat st F is with_the_same_dom holds
dom ((Partial_Sums F) . n) = dom (F . 0)

proof end;

theorem Th12: :: MESFUN9C:12

for X being non empty set
for F being Functional_Sequence of X,REAL
for n being Nat
for x being Element of X
for D being set st F is with_the_same_dom & D c= dom (F . 0) & x in D holds
(Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n

proof end;

theorem Th13: :: MESFUN9C:13

for X being non empty set
for F being Functional_Sequence of X,REAL
for x being Element of X
for D being set st F is with_the_same_dom & D c= dom (F . 0) & x in D holds
( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )

proof end;

theorem Th14: :: MESFUN9C:14

for X being non empty set
for F being Functional_Sequence of X,REAL
for f being PartFunc of X,REAL
for x being Element of X st F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & f . x = Sum (F # x) holds
f . x = lim ((Partial_Sums F) # x)

proof end;

theorem Th15: :: MESFUN9C:15

for X being non empty set
for S being SigmaField of X
for F being Functional_Sequence of X,REAL
for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds
(Partial_Sums F) . n is_simple_func_in S

proof end;

theorem Th16: :: MESFUN9C:16

for X being non empty set
for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,REAL
for m being Nat st ( for n being Nat holds F . n is_measurable_on E ) holds
(Partial_Sums F) . m is_measurable_on E

proof end;

theorem Th17: :: MESFUN9C:17

for X being non empty set
for F being Functional_Sequence of X,REAL st F is with_the_same_dom holds
Partial_Sums F is with_the_same_dom

proof end;

theorem Th18: :: MESFUN9C:18

for X being non empty set
for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,REAL st dom (F . 0) = E & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds
F # x is summable ) holds
lim (Partial_Sums F) is_measurable_on E

proof end;

theorem Th19: :: MESFUN9C:19

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for F being Functional_Sequence of X,REAL st ( for n being Nat holds F . n is_integrable_on M ) holds
for m being Nat holds (Partial_Sums F) . m is_integrable_on M

proof end;

begin

theorem Th20: :: MESFUN9C:20

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

proof end;

Lm1: for X being non empty set
for D being set
for F being Functional_Sequence of X,COMPLEX
for n being Nat holds
( (Re (F || D)) . n = ((Re F) . n) | D & (Im (F || D)) . n = ((Im F) . n) | D )

proof end;

theorem Th21: :: MESFUN9C:21

for X being non empty set
for D being set
for F being Functional_Sequence of X,COMPLEX holds Re (F || D) = (Re F) || D

proof end;

theorem Th22: :: MESFUN9C:22

for X being non empty set
for D being set
for F being Functional_Sequence of X,COMPLEX holds Im (F || D) = (Im F) || D

proof end;

theorem Th23: :: MESFUN9C:23

for X being non empty set
for x being Element of X
for D being set
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom & D c= dom (F . 0) & x in D & F # x is convergent holds
(F || D) # x is convergent

proof end;

theorem Th24: :: MESFUN9C:24

for X being non empty set
for F being Functional_Sequence of X,COMPLEX holds
( F is with_the_same_dom iff Re F is with_the_same_dom )

proof end;

theorem Th25: :: MESFUN9C:25

for X being non empty set
for F being Functional_Sequence of X,COMPLEX holds
( Re F is with_the_same_dom iff Im F is with_the_same_dom )

proof end;

theorem :: MESFUN9C:26

for X being non empty set
for D being set
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom & D = dom (F . 0) & ( for x being Element of X st x in D holds
F # x is convergent ) holds
(lim F) | D = lim (F || D)

proof end;

theorem :: MESFUN9C:27

for X being non empty set
for S being SigmaField of X
for E being Element of S
for n being Nat
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds F . m is_measurable_on E ) holds
(F || E) . n is_measurable_on E

proof end;

theorem :: MESFUN9C:28

for X being non empty set
for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,COMPLEX st E c= dom (F . 0) & F is with_the_same_dom & ( for x being Element of X st x in E holds
F # x is summable ) holds
for x being Element of X st x in E holds
(F || E) # x is summable

proof end;

definition

let X be non empty set ;
let F be Functional_Sequence of X,COMPLEX;
func Partial_Sums F -> Functional_Sequence of X,COMPLEX means :Def3: :: MESFUN9C:def 3
( it . 0 = F . 0 & ( for n being Nat holds it . (n + 1) = (it . n) + (F . (n + 1)) ) );
existence
ex b₁ being Functional_Sequence of X,COMPLEX st
( b₁ . 0 = F . 0 & ( for n being Nat holds b₁ . (n + 1) = (b₁ . n) + (F . (n + 1)) ) )

proof end;

uniqueness
for b₁, b₂ being Functional_Sequence of X,COMPLEX st b₁ . 0 = F . 0 & ( for n being Nat holds b₁ . (n + 1) = (b₁ . n) + (F . (n + 1)) ) & b₂ . 0 = F . 0 & ( for n being Nat holds b₂ . (n + 1) = (b₂ . n) + (F . (n + 1)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines Partial_Sums MESFUN9C:def 3 :
for X being non empty set
for F, b₃ being Functional_Sequence of X,COMPLEX holds
( b₃ = Partial_Sums F iff ( b₃ . 0 = F . 0 & ( for n being Nat holds b₃ . (n + 1) = (b₃ . n) + (F . (n + 1)) ) ) );

theorem Th29: :: MESFUN9C:29

for X being non empty set
for F being Functional_Sequence of X,COMPLEX holds
( Partial_Sums (Re F) = Re (Partial_Sums F) & Partial_Sums (Im F) = Im (Partial_Sums F) )

proof end;

theorem :: MESFUN9C:30

for X being non empty set
for n, m being Nat
for z being set
for F being Functional_Sequence of X,COMPLEX st z in dom ((Partial_Sums F) . n) & m <= n holds
( z in dom ((Partial_Sums F) . m) & z in dom (F . m) )

proof end;

theorem :: MESFUN9C:31

for X being non empty set
for n being Nat
for F being Functional_Sequence of X,COMPLEX holds dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n }

proof end;

theorem Th32: :: MESFUN9C:32

for X being non empty set
for n being Nat
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom holds
dom ((Partial_Sums F) . n) = dom (F . 0)

proof end;

theorem :: MESFUN9C:33

for X being non empty set
for n being Nat
for x being Element of X
for D being set
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom & D c= dom (F . 0) & x in D holds
(Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n

proof end;

theorem Th34: :: MESFUN9C:34

for X being non empty set
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom holds
Partial_Sums F is with_the_same_dom

proof end;

theorem Th35: :: MESFUN9C:35

for X being non empty set
for x being Element of X
for D being set
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom & D c= dom (F . 0) & x in D holds
( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )

proof end;

theorem :: MESFUN9C:36

for X being non empty set
for x being Element of X
for F being Functional_Sequence of X,COMPLEX
for f being PartFunc of X,COMPLEX st F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & F # x is summable & f . x = Sum (F # x) holds
f . x = lim ((Partial_Sums F) # x)

proof end;

theorem :: MESFUN9C:37

for X being non empty set
for S being SigmaField of X
for n being Nat
for F being Functional_Sequence of X,COMPLEX st ( for m being Nat holds F . m is_simple_func_in S ) holds
(Partial_Sums F) . n is_simple_func_in S

proof end;

Lm2: for X being non empty set
for S being SigmaField of X
for E being Element of S
for m being Nat
for F being Functional_Sequence of X,COMPLEX st ( for n being Nat holds F . n is_measurable_on E ) holds
( (Re F) . m is_measurable_on E & (Im F) . m is_measurable_on E )

proof end;

theorem :: MESFUN9C:38

for X being non empty set
for S being SigmaField of X
for E being Element of S
for m being Nat
for F being Functional_Sequence of X,COMPLEX st ( for n being Nat holds F . n is_measurable_on E ) holds
(Partial_Sums F) . m is_measurable_on E

proof end;

theorem :: MESFUN9C:39

for X being non empty set
for S being SigmaField of X
for E being Element of S
for F being Functional_Sequence of X,COMPLEX st dom (F . 0) = E & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds
F # x is summable ) holds
lim (Partial_Sums F) is_measurable_on E

proof end;

theorem :: MESFUN9C:40

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for F being Functional_Sequence of X,COMPLEX st ( for n being Nat holds F . n is_integrable_on M ) holds
for m being Nat holds (Partial_Sums F) . m is_integrable_on M

proof end;

begin

theorem :: MESFUN9C:41

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_simple_func_in S holds
f is_measurable_on A

proof end;

theorem :: MESFUN9C:42

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_simple_func_in S holds
f | A is_simple_func_in S

proof end;

theorem :: MESFUN9C:43

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
dom f is Element of S

proof end;

theorem :: MESFUN9C:44

for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX st f is_simple_func_in S & g is_simple_func_in S holds
f + g is_simple_func_in S

proof end;

theorem :: MESFUN9C:45

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for c being complex number st f is_simple_func_in S holds
c (#) f is_simple_func_in S

proof end;

begin

theorem Th46: :: MESFUN9C:46

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,ExtREAL
for P being PartFunc of X,ExtREAL st E = dom (F . 0) & E = dom P & ( for n being Nat holds F . n is_measurable_on E ) & P is_integrable_on M & ( for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x ) & ( for x being Element of X st x in E holds
F # x is convergent ) holds
lim F is_integrable_on M

proof end;

theorem Th47: :: MESFUN9C:47

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,REAL
for P being PartFunc of X,REAL st E = dom (F . 0) & E = dom P & ( for n being Nat holds F . n is_measurable_on E ) & P is_integrable_on M & ( for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x ) & ( for x being Element of X st x in E holds
F # x is convergent ) holds
lim F is_integrable_on M

proof end;

theorem Th48: :: MESFUN9C:48

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,REAL
for P being PartFunc of X,REAL st E = dom (F . 0) & E = dom P & ( for n being Nat holds F . n is_measurable_on E ) & P is_integrable_on M & ( for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x ) holds
ex I being Real_Sequence st
( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & ( ( for x being Element of X st x in E holds
F # x is convergent ) implies ( I is convergent & lim I = Integral (M,(lim F)) ) ) )

proof end;

definition

let X be set ;
let F be Functional_Sequence of X,REAL;
attr F is uniformly_bounded means :Def4: :: MESFUN9C:def 4
ex K being real number st
for n being Nat
for x being Element of X st x in dom (F . 0) holds
|.((F . n) . x).| <= K;

end;

:: deftheorem Def4 defines uniformly_bounded MESFUN9C:def 4 :
for X being set
for F being Functional_Sequence of X,REAL holds
( F is uniformly_bounded iff ex K being real number st
for n being Nat
for x being Element of X st x in dom (F . 0) holds
|.((F . n) . x).| <= K );

theorem Th49: :: MESFUN9C:49

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,REAL st M . E < +infty & E = dom (F . 0) & ( for n being Nat holds F . n is_measurable_on E ) & F is uniformly_bounded & ( for x being Element of X st x in E holds
F # x is convergent ) holds
( ( for n being Nat holds F . n is_integrable_on M ) & lim F is_integrable_on M & ex I being ExtREAL_sequence st
( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & lim I = Integral (M,(lim F)) ) )

proof end;

definition

let X be set ;
let F be Functional_Sequence of X,REAL;
let f be PartFunc of X,REAL;
pred F is_uniformly_convergent_to f means :Def5: :: MESFUN9C:def 5
( F is with_the_same_dom & dom (F . 0) = dom f & ( for e being real number st e > 0 holds
ex N being Nat st
for n being Nat
for x being Element of X st n >= N & x in dom (F . 0) holds
|.(((F . n) . x) - (f . x)).| < e ) );

end;

:: deftheorem Def5 defines is_uniformly_convergent_to MESFUN9C:def 5 :
for X being set
for F being Functional_Sequence of X,REAL
for f being PartFunc of X,REAL holds
( F is_uniformly_convergent_to f iff ( F is with_the_same_dom & dom (F . 0) = dom f & ( for e being real number st e > 0 holds
ex N being Nat st
for n being Nat
for x being Element of X st n >= N & x in dom (F . 0) holds
|.(((F . n) . x) - (f . x)).| < e ) ) );

theorem Th50: :: MESFUN9C:50

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,REAL
for f being PartFunc of X,REAL st M . E < +infty & E = dom (F . 0) & ( for n being Nat holds F . n is_integrable_on M ) & F is_uniformly_convergent_to f holds
( f is_integrable_on M & ex I being ExtREAL_sequence st
( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & lim I = Integral (M,f) ) )

proof end;

theorem Th51: :: MESFUN9C:51

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for P being PartFunc of X,REAL
for F being with_the_same_dom Functional_Sequence of X,COMPLEX st E = dom (F . 0) & E = dom P & ( for n being Nat holds F . n is_measurable_on E ) & P is_integrable_on M & ( for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x ) & ( for x being Element of X st x in E holds
F # x is convergent ) holds
lim F is_integrable_on M

proof end;

theorem :: MESFUN9C:52

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for P being PartFunc of X,REAL
for F being with_the_same_dom Functional_Sequence of X,COMPLEX st E = dom (F . 0) & E = dom P & ( for n being Nat holds F . n is_measurable_on E ) & P is_integrable_on M & ( for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x ) holds
ex I being Complex_Sequence st
( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & ( ( for x being Element of X st x in E holds
F # x is convergent ) implies ( I is convergent & lim I = Integral (M,(lim F)) ) ) )

proof end;

definition

let X be set ;
let F be Functional_Sequence of X,COMPLEX;
attr F is uniformly_bounded means :Def6: :: MESFUN9C:def 6
ex K being real number st
for n being Nat
for x being Element of X st x in dom (F . 0) holds
|.((F . n) . x).| <= K;

end;

:: deftheorem Def6 defines uniformly_bounded MESFUN9C:def 6 :
for X being set
for F being Functional_Sequence of X,COMPLEX holds
( F is uniformly_bounded iff ex K being real number st
for n being Nat
for x being Element of X st x in dom (F . 0) holds
|.((F . n) . x).| <= K );

theorem :: MESFUN9C:53

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,COMPLEX st M . E < +infty & E = dom (F . 0) & ( for n being Nat holds F . n is_measurable_on E ) & F is uniformly_bounded & ( for x being Element of X st x in E holds
F # x is convergent ) holds
( ( for n being Nat holds F . n is_integrable_on M ) & lim F is_integrable_on M & ex I being Complex_Sequence st
( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & lim I = Integral (M,(lim F)) ) )

proof end;

definition

let X be set ;
let F be Functional_Sequence of X,COMPLEX;
let f be PartFunc of X,COMPLEX;
pred F is_uniformly_convergent_to f means :Def7: :: MESFUN9C:def 7
( F is with_the_same_dom & dom (F . 0) = dom f & ( for e being real number st e > 0 holds
ex N being Nat st
for n being Nat
for x being Element of X st n >= N & x in dom (F . 0) holds
|.(((F . n) . x) - (f . x)).| < e ) );

end;

:: deftheorem Def7 defines is_uniformly_convergent_to MESFUN9C:def 7 :
for X being set
for F being Functional_Sequence of X,COMPLEX
for f being PartFunc of X,COMPLEX holds
( F is_uniformly_convergent_to f iff ( F is with_the_same_dom & dom (F . 0) = dom f & ( for e being real number st e > 0 holds
ex N being Nat st
for n being Nat
for x being Element of X st n >= N & x in dom (F . 0) holds
|.(((F . n) . x) - (f . x)).| < e ) ) );

theorem :: MESFUN9C:54

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for F being with_the_same_dom Functional_Sequence of X,COMPLEX
for f being PartFunc of X,COMPLEX st M . E < +infty & E = dom (F . 0) & ( for n being Nat holds F . n is_integrable_on M ) & F is_uniformly_convergent_to f holds
( f is_integrable_on M & ex I being Complex_Sequence st
( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & lim I = Integral (M,f) ) )

proof end;