:: MESFUNC5 semantic presentation

theorem Th1: :: MESFUNC5:1
for b1, b2 being R_eal holds |.(b1 - b2).| = |.(b2 - b1).|
proof end;

theorem Th2: :: MESFUNC5:2
for b1, b2 being R_eal holds b2 - b1 <= |.(b1 - b2).|
proof end;

theorem Th3: :: MESFUNC5:3
for b1, b2 being R_eal
for b3 being real number holds
( not |.(b1 - b2).| < b3 or ( b1 = +infty & b2 = +infty ) or ( b1 = -infty & b2 = -infty ) or ( b1 <> +infty & b1 <> -infty & b2 <> +infty & b2 <> -infty ) )
proof end;

theorem Th4: :: MESFUNC5:4
for b1, b2 being R_eal st ( for b3 being real number st 0 < b3 holds
b1 < b2 + (R_EAL b3) ) holds
b1 <= b2
proof end;

theorem Th5: :: MESFUNC5:5
for b1, b2, b3 being R_eal st b3 <> -infty & b3 <> +infty & b1 < b2 holds
b1 + b3 < b2 + b3
proof end;

theorem Th6: :: MESFUNC5:6
for b1, b2, b3 being R_eal st b3 <> -infty & b3 <> +infty & b1 < b2 holds
b1 - b3 < b2 - b3
proof end;

theorem Th7: :: MESFUNC5:7
for b1, b2 being real number holds
( (R_EAL b1) + (R_EAL b2) = b1 + b2 & - (R_EAL b1) = - b1 )
proof end;

theorem Th8: :: MESFUNC5:8
for b1 being Nat
for b2 being R_eal st 0 <= b2 & b2 < b1 holds
ex b3 being Nat st
( 1 <= b3 & b3 <= (2 |^ b1) * b1 & (b3 - 1) / (2 |^ b1) <= b2 & b2 < b3 / (2 |^ b1) )
proof end;

theorem Th9: :: MESFUNC5:9
for b1, b2 being Nat
for b3 being R_eal st 1 <= b2 & b2 <= (2 |^ b1) * b1 & b1 <= b3 & (b2 - 1) / (2 |^ b1) <= b3 holds
b2 / (2 |^ b1) <= b3
proof end;

theorem Th10: :: MESFUNC5:10
for b1, b2, b3, b4 being R_eal st -infty < b3 & b1 < b2 & b3 < b4 holds
b1 + b3 < b2 + b4
proof end;

theorem Th11: :: MESFUNC5:11
for b1, b2, b3 being R_eal st 0 <= b3 holds
( b3 * (max b1,b2) = max (b3 * b1),(b3 * b2) & b3 * (min b1,b2) = min (b3 * b1),(b3 * b2) )
proof end;

theorem Th12: :: MESFUNC5:12
for b1, b2, b3 being R_eal st b3 <= 0 holds
( b3 * (min b1,b2) = max (b3 * b1),(b3 * b2) & b3 * (max b1,b2) = min (b3 * b1),(b3 * b2) )
proof end;

theorem Th13: :: MESFUNC5:13
for b1, b2, b3 being R_eal st 0 <= b1 & 0 <= b3 & b3 + b1 <= b2 holds
b3 <= b2
proof end;

definition
let c1 be set ;
attr a1 is nonpositive means :Def1: :: MESFUNC5:def 1
for b1 being R_eal st b1 in a1 holds
b1 <= 0;
end;

:: deftheorem Def1 defines nonpositive MESFUNC5:def 1 :
for b1 being set holds
( b1 is nonpositive iff for b2 being R_eal st b2 in b1 holds
b2 <= 0 );

definition
let c1 be Relation;
attr a1 is nonpositive means :Def2: :: MESFUNC5:def 2
rng a1 is nonpositive;
end;

:: deftheorem Def2 defines nonpositive MESFUNC5:def 2 :
for b1 being Relation holds
( b1 is nonpositive iff rng b1 is nonpositive );

theorem Th14: :: MESFUNC5:14
for b1 being set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is nonpositive iff for b3 being set holds b2 . b3 <= 0. )
proof end;

theorem Th15: :: MESFUNC5:15
for b1 being set
for b2 being PartFunc of b1, ExtREAL st ( for b3 being set st b3 in dom b2 holds
b2 . b3 <= 0. ) holds
b2 is nonpositive
proof end;

definition
let c1 be Relation;
attr a1 is without-infty means :Def3: :: MESFUNC5:def 3
not -infty in rng a1;
attr a1 is without+infty means :Def4: :: MESFUNC5:def 4
not +infty in rng a1;
end;

:: deftheorem Def3 defines without-infty MESFUNC5:def 3 :
for b1 being Relation holds
( b1 is without-infty iff not -infty in rng b1 );

:: deftheorem Def4 defines without+infty MESFUNC5:def 4 :
for b1 being Relation holds
( b1 is without+infty iff not +infty in rng b1 );

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
redefine attr without-infty as a2 is without-infty means :Def5: :: MESFUNC5:def 5
for b1 being set holds -infty < a2 . b1;
compatibility
( c2 is without-infty iff for b1 being set holds -infty < c2 . b1 )
proof end;
redefine attr without+infty as a2 is without+infty means :Def6: :: MESFUNC5:def 6
for b1 being set holds a2 . b1 < +infty ;
compatibility
( c2 is without+infty iff for b1 being set holds c2 . b1 < +infty )
proof end;
end;

:: deftheorem Def5 defines without-infty MESFUNC5:def 5 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is without-infty iff for b3 being set holds -infty < b2 . b3 );

:: deftheorem Def6 defines without+infty MESFUNC5:def 6 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is without+infty iff for b3 being set holds b2 . b3 < +infty );

theorem Th16: :: MESFUNC5:16
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( ( for b3 being set st b3 in dom b2 holds
-infty < b2 . b3 ) iff b2 is without-infty )
proof end;

theorem Th17: :: MESFUNC5:17
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( ( for b3 being set st b3 in dom b2 holds
b2 . b3 < +infty ) iff b2 is without+infty )
proof end;

theorem Th18: :: MESFUNC5:18
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL st b2 is nonnegative holds
b2 is without-infty
proof end;

theorem Th19: :: MESFUNC5:19
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL st b2 is nonpositive holds
b2 is without+infty
proof end;

registration
let c1 be non empty set ;
cluster nonnegative -> V174 Relation of a1, ExtREAL ;
coherence
for b1 being PartFunc of c1, ExtREAL st b1 is nonnegative holds
b1 is without-infty
by Th18;
cluster nonpositive -> V175 Relation of a1, ExtREAL ;
coherence
for b1 being PartFunc of c1, ExtREAL st b1 is nonpositive holds
b1 is without+infty
by Th19;
end;

theorem Th20: :: MESFUNC5:20
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL st b3 is_simple_func_in b2 holds
( b3 is without+infty & b3 is without-infty )
proof end;

theorem Th21: :: MESFUNC5:21
for b1 being non empty set
for b2 being set
for b3 being PartFunc of b1, ExtREAL st b3 is nonnegative holds
b3 | b2 is nonnegative
proof end;

theorem Th22: :: MESFUNC5:22
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL st b2 is without-infty & b3 is without-infty holds
dom (b2 + b3) = (dom b2) /\ (dom b3)
proof end;

theorem Th23: :: MESFUNC5:23
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL st b2 is without-infty & b3 is without+infty holds
dom (b2 - b3) = (dom b2) /\ (dom b3)
proof end;

theorem Th24: :: MESFUNC5:24
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Function of RAT ,b2
for b6 being Real
for b7 being Element of b2 st b3 is without-infty & b4 is without-infty & ( for b8 being Rational holds b5 . b8 = (b7 /\ (less_dom b3,(R_EAL b8))) /\ (b7 /\ (less_dom b4,(R_EAL (b6 - b8)))) ) holds
b7 /\ (less_dom (b3 + b4),(R_EAL b6)) = union (rng b5)
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, REAL ;
func R_EAL c2 -> PartFunc of a1, ExtREAL equals :: MESFUNC5:def 7
a2;
coherence
c2 is PartFunc of c1, ExtREAL
by PARTFUN1:31;
end;

:: deftheorem Def7 defines R_EAL MESFUNC5:def 7 :
for b1 being non empty set
for b2 being PartFunc of b1, REAL holds R_EAL b2 = b2;

theorem Th25: :: MESFUNC5:25
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is nonnegative & b5 is nonnegative holds
b4 + b5 is nonnegative
proof end;

theorem Th26: :: MESFUNC5:26
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real st b2 is nonnegative holds
( ( 0 <= b3 implies b3 (#) b2 is nonnegative ) & ( b3 <= 0 implies b3 (#) b2 is nonpositive ) )
proof end;

theorem Th27: :: MESFUNC5:27
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL st ( for b4 being set st b4 in (dom b2) /\ (dom b3) holds
( b3 . b4 <= b2 . b4 & -infty < b3 . b4 & b2 . b4 < +infty ) ) holds
b2 - b3 is nonnegative
proof end;

Lemma31: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( max+ b4 is nonnegative & max- b4 is nonnegative & |.b4.| is nonnegative )
proof end;

theorem Th28: :: MESFUNC5:28
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL st b2 is nonnegative & b3 is nonnegative holds
( dom (b2 + b3) = (dom b2) /\ (dom b3) & b2 + b3 is nonnegative )
proof end;

theorem Th29: :: MESFUNC5:29
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, ExtREAL st b2 is nonnegative & b3 is nonnegative & b4 is nonnegative holds
( dom ((b2 + b3) + b4) = ((dom b2) /\ (dom b3)) /\ (dom b4) & (b2 + b3) + b4 is nonnegative & ( for b5 being set st b5 in ((dom b2) /\ (dom b3)) /\ (dom b4) holds
((b2 + b3) + b4) . b5 = ((b2 . b5) + (b3 . b5)) + (b4 . b5) ) )
proof end;

theorem Th30: :: MESFUNC5:30
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL st b2 is without-infty & b3 is without-infty holds
( dom ((max+ (b2 + b3)) + (max- b2)) = (dom b2) /\ (dom b3) & dom ((max- (b2 + b3)) + (max+ b2)) = (dom b2) /\ (dom b3) & dom (((max+ (b2 + b3)) + (max- b2)) + (max- b3)) = (dom b2) /\ (dom b3) & dom (((max- (b2 + b3)) + (max+ b2)) + (max+ b3)) = (dom b2) /\ (dom b3) & (max+ (b2 + b3)) + (max- b2) is nonnegative & (max- (b2 + b3)) + (max+ b2) is nonnegative )
proof end;

theorem Th31: :: MESFUNC5:31
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL st b2 is without-infty & b2 is without+infty & b3 is without-infty & b3 is without+infty holds
((max+ (b2 + b3)) + (max- b2)) + (max- b3) = ((max- (b2 + b3)) + (max+ b2)) + (max+ b3)
proof end;

theorem Th32: :: MESFUNC5:32
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real st 0 <= b3 holds
( max+ (b3 (#) b2) = b3 (#) (max+ b2) & max- (b3 (#) b2) = b3 (#) (max- b2) )
proof end;

theorem Th33: :: MESFUNC5:33
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real st 0 <= b3 holds
( max+ ((- b3) (#) b2) = b3 (#) (max- b2) & max- ((- b3) (#) b2) = b3 (#) (max+ b2) )
proof end;

theorem Th34: :: MESFUNC5:34
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set holds
( max+ (b2 | b3) = (max+ b2) | b3 & max- (b2 | b3) = (max- b2) | b3 )
proof end;

theorem Th35: :: MESFUNC5:35
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL
for b4 being set st b4 c= dom (b2 + b3) holds
( dom ((b2 + b3) | b4) = b4 & dom ((b2 | b4) + (b3 | b4)) = b4 & (b2 + b3) | b4 = (b2 | b4) + (b3 | b4) )
proof end;

theorem Th36: :: MESFUNC5:36
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal holds eq_dom b2,b3 = b2 " {b3}
proof end;

theorem Th37: :: MESFUNC5:37
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st b3 is without-infty & b4 is without-infty & b3 is_measurable_on b5 & b4 is_measurable_on b5 holds
b3 + b4 is_measurable_on b5
proof end;

Lemma42: for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st b3 is nonnegative & b4 is nonnegative & b3 is_measurable_on b5 & b4 is_measurable_on b5 holds
( dom (b3 + b4) = (dom b3) /\ (dom b4) & b3 + b4 is_measurable_on b5 )
proof end;

theorem Th38: :: MESFUNC5:38
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & dom b4 = {} holds
ex b5 being Finite_Sep_Sequence of b2ex b6, b7 being FinSequence of ExtREAL st
( b5,b6 are_Re-presentation_of b4 & b6 . 1 = 0 & ( for b8 being Nat st 2 <= b8 & b8 in dom b6 holds
( 0 < b6 . b8 & b6 . b8 < +infty ) ) & dom b7 = dom b5 & ( for b8 being Nat st b8 in dom b7 holds
b7 . b8 = (b6 . b8) * ((b3 * b5) . b8) ) & Sum b7 = 0 )
proof end;

theorem Th39: :: MESFUNC5:39
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2
for b5, b6 being Real st b3 is_measurable_on b4 & b4 c= dom b3 holds
(b4 /\ (great_eq_dom b3,(R_EAL b5))) /\ (less_dom b3,(R_EAL b6)) is_measurable_on b2
proof end;

theorem Th40: :: MESFUNC5:40
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st b4 is_simple_func_in b2 holds
b4 | b5 is_simple_func_in b2
proof end;

theorem Th41: :: MESFUNC5:41
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Element of b2
for b4 being Finite_Sep_Sequence of b2
for b5 being FinSequence st dom b4 = dom b5 & ( for b6 being Nat st b6 in dom b4 holds
b5 . b6 = (b4 . b6) /\ b3 ) holds
b5 is Finite_Sep_Sequence of b2
proof end;

theorem Th42: :: MESFUNC5:42
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2
for b5, b6 being Finite_Sep_Sequence of b2
for b7 being FinSequence of ExtREAL st dom b5 = dom b6 & ( for b8 being Nat st b8 in dom b5 holds
b6 . b8 = (b5 . b8) /\ b4 ) & b5,b7 are_Re-presentation_of b3 holds
b6,b7 are_Re-presentation_of b3 | b4
proof end;

theorem Th43: :: MESFUNC5:43
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 holds
dom b4 is Element of b2
proof end;

Lemma48: for b1 being set
for b2 being FinSequence st ( for b3 being Nat st b3 in dom b2 holds
b2 . b3 in b1 ) holds
b2 is FinSequence of b1
proof end;

Lemma49: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & dom b4 <> {} & b5 is_simple_func_in b2 & dom b5 = dom b4 holds
( b4 + b5 is_simple_func_in b2 & dom (b4 + b5) <> {} )
proof end;

theorem Th44: :: MESFUNC5:44
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL st b3 is_simple_func_in b2 & b4 is_simple_func_in b2 holds
b3 + b4 is_simple_func_in b2
proof end;

theorem Th45: :: MESFUNC5:45
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st b4 is_simple_func_in b2 holds
b5 (#) b4 is_simple_func_in b2
proof end;

theorem Th46: :: MESFUNC5:46
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & ( for b6 being set st b6 in dom (b4 - b5) holds
b5 . b6 <= b4 . b6 ) holds
b4 - b5 is nonnegative
proof end;

theorem Th47: :: MESFUNC5:47
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5 being R_eal st b5 <> +infty & b5 <> -infty holds
ex b6 being PartFunc of b1, ExtREAL st
( b6 is_simple_func_in b2 & dom b6 = b4 & ( for b7 being set st b7 in b4 holds
b6 . b7 = b5 ) )
proof end;

theorem Th48: :: MESFUNC5:48
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st b4 is_measurable_on b5 & b6 = (dom b4) /\ b5 holds
b4 | b5 is_measurable_on b6
proof end;

theorem Th49: :: MESFUNC5:49
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being PartFunc of b1, ExtREAL st b4 c= dom b5 & b5 is_measurable_on b4 & b6 is_measurable_on b4 & b5 is without-infty & b6 is without-infty holds
(max+ (b5 + b6)) + (max- b5) is_measurable_on b4
proof end;

theorem Th50: :: MESFUNC5:50
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being PartFunc of b1, ExtREAL st b4 c= (dom b5) /\ (dom b6) & b5 is_measurable_on b4 & b6 is_measurable_on b4 & b5 is without-infty & b6 is without-infty holds
(max- (b5 + b6)) + (max+ b5) is_measurable_on b4
proof end;

theorem Th51: :: MESFUNC5:51
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being set st b4 in b2 holds
0 <= b3 . b4
proof end;

Lemma58: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st dom b4 in b2 & ( for b6 being set st b6 in dom b4 holds
b4 . b6 = b5 ) holds
b4 is_simple_func_in b2
proof end;

theorem Th52: :: MESFUNC5:52
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & ex b6 being Element of b2 st
( b6 = dom b5 & b5 is_measurable_on b6 ) & b4 " {+infty } in b2 & b4 " {-infty } in b2 & b5 " {+infty } in b2 & b5 " {-infty } in b2 holds
dom (b4 + b5) in b2
proof end;

Lemma60: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Element of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being real number holds b3 /\ (less_dom b4,(R_EAL b5)) = less_dom (b4 | b3),(R_EAL b5)
proof end;

Lemma61: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5 being PartFunc of b1, ExtREAL holds
( b5 | b4 is_measurable_on b4 iff b5 is_measurable_on b4 )
proof end;

Lemma62: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & ex b6 being Element of b2 st
( b6 = dom b5 & b5 is_measurable_on b6 ) & dom b4 = dom b5 holds
ex b6 being Element of b2 st
( b6 = dom (b4 + b5) & b4 + b5 is_measurable_on b6 )
proof end;

theorem Th53: :: MESFUNC5:53
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & ex b6 being Element of b2 st
( b6 = dom b5 & b5 is_measurable_on b6 ) holds
ex b6 being Element of b2 st
( b6 = dom (b4 + b5) & b4 + b5 is_measurable_on b6 )
proof end;

theorem Th54: :: MESFUNC5:54
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st dom b4 = b5 holds
( b4 is_measurable_on b6 iff b4 is_measurable_on b5 /\ b6 )
proof end;

theorem Th55: :: MESFUNC5:55
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st dom b4 = b5 holds
for b5 being Real
for b6 being Element of b2 st b4 is_measurable_on b6 holds
b5 (#) b4 is_measurable_on b6
proof end;

definition
mode ExtREAL_sequence is Function of NAT , ExtREAL ;
end;

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent_to_finite_number means :Def8: :: MESFUNC5:def 8
ex b1 being real number st
for b2 being real number st 0 < b2 holds
ex b3 being Nat st
for b4 being Nat st b3 <= b4 holds
|.((a1 . b4) - (R_EAL b1)).| < b2;
end;

:: deftheorem Def8 defines convergent_to_finite_number MESFUNC5:def 8 :
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_finite_number iff ex b2 being real number st
for b3 being real number st 0 < b3 holds
ex b4 being Nat st
for b5 being Nat st b4 <= b5 holds
|.((b1 . b5) - (R_EAL b2)).| < b3 );

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent_to_+infty means :Def9: :: MESFUNC5:def 9
for b1 being real number st 0 < b1 holds
ex b2 being Nat st
for b3 being Nat st b2 <= b3 holds
b1 <= a1 . b3;
end;

:: deftheorem Def9 defines convergent_to_+infty MESFUNC5:def 9 :
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_+infty iff for b2 being real number st 0 < b2 holds
ex b3 being Nat st
for b4 being Nat st b3 <= b4 holds
b2 <= b1 . b4 );

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent_to_-infty means :Def10: :: MESFUNC5:def 10
for b1 being real number st b1 < 0 holds
ex b2 being Nat st
for b3 being Nat st b2 <= b3 holds
a1 . b3 <= b1;
end;

:: deftheorem Def10 defines convergent_to_-infty MESFUNC5:def 10 :
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_-infty iff for b2 being real number st b2 < 0 holds
ex b3 being Nat st
for b4 being Nat st b3 <= b4 holds
b1 . b4 <= b2 );

theorem Th56: :: MESFUNC5:56
for b1 being ExtREAL_sequence st b1 is convergent_to_+infty holds
( not b1 is convergent_to_-infty & not b1 is convergent_to_finite_number )
proof end;

theorem Th57: :: MESFUNC5:57
for b1 being ExtREAL_sequence st b1 is convergent_to_-infty holds
( not b1 is convergent_to_+infty & not b1 is convergent_to_finite_number )
proof end;

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent means :Def11: :: MESFUNC5:def 11
( a1 is convergent_to_finite_number or a1 is convergent_to_+infty or a1 is convergent_to_-infty );
end;

:: deftheorem Def11 defines convergent MESFUNC5:def 11 :
for b1 being ExtREAL_sequence holds
( b1 is convergent iff ( b1 is convergent_to_finite_number or b1 is convergent_to_+infty or b1 is convergent_to_-infty ) );

definition
let c1 be ExtREAL_sequence;
assume E72: c1 is convergent ;
func lim c1 -> R_eal means :Def12: :: MESFUNC5:def 12
( ex b1 being real number st
( a2 = b1 & ( for b2 being real number st 0 < b2 holds
ex b3 being Nat st
for b4 being Nat st b3 <= b4 holds
|.((a1 . b4) - a2).| < b2 ) & a1 is convergent_to_finite_number ) or ( a2 = +infty & a1 is convergent_to_+infty ) or ( a2 = -infty & a1 is convergent_to_-infty ) );
existence
ex b1 being R_eal st
( ex b2 being real number st
( b1 = b2 & ( for b3 being real number st 0 < b3 holds
ex b4 being Nat st
for b5 being Nat st b4 <= b5 holds
|.((c1 . b5) - b1).| < b3 ) & c1 is convergent_to_finite_number ) or ( b1 = +infty & c1 is convergent_to_+infty ) or ( b1 = -infty & c1 is convergent_to_-infty ) )
proof end;
uniqueness
for b1, b2 being R_eal st ( ex b3 being real number st
( b1 = b3 & ( for b4 being real number st 0 < b4 holds
ex b5 being Nat st
for b6 being Nat st b5 <= b6 holds
|.((c1 . b6) - b1).| < b4 ) & c1 is convergent_to_finite_number ) or ( b1 = +infty & c1 is convergent_to_+infty ) or ( b1 = -infty & c1 is convergent_to_-infty ) ) & ( ex b3 being real number st
( b2 = b3 & ( for b4 being real number st 0 < b4 holds
ex b5 being Nat st
for b6 being Nat st b5 <= b6 holds
|.((c1 . b6) - b2).| < b4 ) & c1 is convergent_to_finite_number ) or ( b2 = +infty & c1 is convergent_to_+infty ) or ( b2 = -infty & c1 is convergent_to_-infty ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines lim MESFUNC5:def 12 :
for b1 being ExtREAL_sequence st b1 is convergent holds
for b2 being R_eal holds
( b2 = lim b1 iff ( ex b3 being real number st
( b2 = b3 & ( for b4 being real number st 0 < b4 holds
ex b5 being Nat st
for b6 being Nat st b5 <= b6 holds
|.((b1 . b6) - b2).| < b4 ) & b1 is convergent_to_finite_number ) or ( b2 = +infty & b1 is convergent_to_+infty ) or ( b2 = -infty & b1 is convergent_to_-infty ) ) );

theorem Th58: :: MESFUNC5:58
for b1 being ExtREAL_sequence
for b2 being real number st ( for b3 being Nat holds b1 . b3 = b2 ) holds
( b1 is convergent_to_finite_number & lim b1 = b2 )
proof end;

theorem Th59: :: MESFUNC5:59
for b1 being FinSequence of ExtREAL st ( for b2 being Nat st b2 in dom b1 holds
0 <= b1 . b2 ) holds
0 <= Sum b1
proof end;

theorem Th60: :: MESFUNC5:60
for b1 being ExtREAL_sequence st ( for b2, b3 being Nat st b2 <= b3 holds
b1 . b2 <= b1 . b3 ) holds
( b1 is convergent & lim b1 = sup (rng b1) )
proof end;

theorem Th61: :: MESFUNC5:61
for b1, b2 being ExtREAL_sequence st ( for b3 being Nat holds b1 . b3 <= b2 . b3 ) holds
sup (rng b1) <= sup (rng b2)
proof end;

theorem Th62: :: MESFUNC5:62
for b1 being ExtREAL_sequence
for b2 being Nat holds b1 . b2 <= sup (rng b1)
proof end;

theorem Th63: :: MESFUNC5:63
for b1 being ExtREAL_sequence
for b2 being R_eal st ( for b3 being Nat holds b1 . b3 <= b2 ) holds
sup (rng b1) <= b2
proof end;

theorem Th64: :: MESFUNC5:64
for b1 being ExtREAL_sequence
for b2 being R_eal st b2 <> +infty & ( for b3 being Nat holds b1 . b3 <= b2 ) holds
sup (rng b1) < +infty
proof end;

theorem Th65: :: MESFUNC5:65
for b1 being ExtREAL_sequence st b1 is without-infty holds
( sup (rng b1) <> +infty iff ex b2 being real number st
( 0 < b2 & ( for b3 being Nat holds b1 . b3 <= b2 ) ) )
proof end;

theorem Th66: :: MESFUNC5:66
for b1 being ExtREAL_sequence
for b2 being R_eal st ( for b3 being Nat holds b1 . b3 = b2 ) holds
( b1 is convergent & lim b1 = b2 & lim b1 = sup (rng b1) )
proof end;

Lemma81: for b1 being ExtREAL_sequence holds
( not b1 is without-infty or sup (rng b1) in REAL or sup (rng b1) = +infty )
proof end;

theorem Th67: :: MESFUNC5:67
for b1, b2, b3 being ExtREAL_sequence st ( for b4, b5 being Nat st b4 <= b5 holds
b1 . b4 <= b1 . b5 ) & ( for b4, b5 being Nat st b4 <= b5 holds
b2 . b4 <= b2 . b5 ) & b1 is without-infty & b2 is without-infty & ( for b4 being Nat holds (b1 . b4) + (b2 . b4) = b3 . b4 ) holds
( b3 is convergent & lim b3 = sup (rng b3) & lim b3 = (lim b1) + (lim b2) & sup (rng b3) = (sup (rng b2)) + (sup (rng b1)) )
proof end;

theorem Th68: :: MESFUNC5:68
for b1, b2 being ExtREAL_sequence
for b3 being Real st 0 <= b3 & b1 is without-infty & ( for b4 being Nat holds b2 . b4 = (R_EAL b3) * (b1 . b4) ) holds
( sup (rng b2) = (R_EAL b3) * (sup (rng b1)) & b2 is without-infty )
proof end;

theorem Th69: :: MESFUNC5:69
for b1, b2 being ExtREAL_sequence
for b3 being Real st 0 <= b3 & ( for b4, b5 being Nat st b4 <= b5 holds
b1 . b4 <= b1 . b5 ) & ( for b4 being Nat holds b2 . b4 = (R_EAL b3) * (b1 . b4) ) & b1 is without-infty holds
( ( for b4, b5 being Nat st b4 <= b5 holds
b2 . b4 <= b2 . b5 ) & b2 is without-infty & b2 is convergent & lim b2 = sup (rng b2) & lim b2 = (R_EAL b3) * (lim b1) )
proof end;

definition
let c1 be non empty set ;
let c2 be Functional_Sequence of c1, ExtREAL ;
let c3 be Element of c1;
func c2 # c3 -> ExtREAL_sequence means :Def13: :: MESFUNC5:def 13
for b1 being Nat holds a4 . b1 = (a2 . b1) . a3;
existence
ex b1 being ExtREAL_sequence st
for b2 being Nat holds b1 . b2 = (c2 . b2) . c3
proof end;
uniqueness
for b1, b2 being ExtREAL_sequence st ( for b3 being Nat holds b1 . b3 = (c2 . b3) . c3 ) & ( for b3 being Nat holds b2 . b3 = (c2 . b3) . c3 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def13 defines # MESFUNC5:def 13 :
for b1 being non empty set
for b2 being Functional_Sequence of b1, ExtREAL
for b3 being Element of b1
for b4 being ExtREAL_sequence holds
( b4 = b2 # b3 iff for b5 being Nat holds b4 . b5 = (b2 . b5) . b3 );

definition
let c1, c2 be set ;
let c3 be Function of NAT , PFuncs c1,c2;
let c4 be Nat;
redefine func . as c3 . c4 -> PartFunc of a1,a2;
coherence
c3 . c4 is PartFunc of c1,c2
proof end;
end;

theorem Th70: :: MESFUNC5:70
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL st ex b4 being Element of b2 st
( b4 = dom b3 & b3 is_measurable_on b4 ) & b3 is nonnegative holds
ex b4 being Functional_Sequence of b1, ExtREAL st
( ( for b5 being Nat holds
( b4 . b5 is_simple_func_in b2 & dom (b4 . b5) = dom b3 ) ) & ( for b5 being Nat holds b4 . b5 is nonnegative ) & ( for b5, b6 being Nat st b5 <= b6 holds
for b7 being Element of b1 st b7 in dom b3 holds
(b4 . b5) . b7 <= (b4 . b6) . b7 ) & ( for b5 being Element of b1 st b5 in dom b3 holds
( b4 # b5 is convergent & lim (b4 # b5) = b3 . b5 ) ) )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
func integral' c3,c4 -> Element of ExtREAL equals :Def14: :: MESFUNC5:def 14
integral a1,a2,a3,a4 if dom a4 <> {}
otherwise 0. ;
correctness
coherence
( ( dom c4 <> {} implies integral c1,c2,c3,c4 is Element of ExtREAL ) & ( not dom c4 <> {} implies 0. is Element of ExtREAL ) )
;
consistency
for b1 being Element of ExtREAL holds verum
;
;
end;

:: deftheorem Def14 defines integral' MESFUNC5:def 14 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ( dom b4 <> {} implies integral' b3,b4 = integral b1,b2,b3,b4 ) & ( not dom b4 <> {} implies integral' b3,b4 = 0. ) );

theorem Th71: :: MESFUNC5:71
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & b4 is nonnegative & b5 is nonnegative holds
( dom (b4 + b5) = (dom b4) /\ (dom b5) & integral' b3,(b4 + b5) = (integral' b3,(b4 | (dom (b4 + b5)))) + (integral' b3,(b5 | (dom (b4 + b5)))) )
proof end;

theorem Th72: :: MESFUNC5:72
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st b4 is_simple_func_in b2 & b4 is nonnegative & 0 <= b5 holds
integral' b3,(b5 (#) b4) = (R_EAL b5) * (integral' b3,b4)
proof end;

theorem Th73: :: MESFUNC5:73
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st b4 is_simple_func_in b2 & b4 is nonnegative & b5 misses b6 holds
integral' b3,(b4 | (b5 \/ b6)) = (integral' b3,(b4 | b5)) + (integral' b3,(b4 | b6))
proof end;

theorem Th74: :: MESFUNC5:74
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b4 is nonnegative holds
0 <= integral' b3,b4
proof end;

Lemma92: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & dom b4 <> {} & b4 is nonnegative & b5 is_simple_func_in b2 & dom b5 = dom b4 & b5 is nonnegative & ( for b6 being set st b6 in dom b4 holds
b5 . b6 <= b4 . b6 ) holds
( b4 - b5 is_simple_func_in b2 & dom (b4 - b5) <> {} & b4 - b5 is nonnegative & integral b1,b2,b3,b4 = (integral b1,b2,b3,(b4 - b5)) + (integral b1,b2,b3,b5) )
proof end;

theorem Th75: :: MESFUNC5:75
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b4 is nonnegative & b5 is_simple_func_in b2 & b5 is nonnegative & ( for b6 being set st b6 in dom (b4 - b5) holds
b5 . b6 <= b4 . b6 ) holds
( dom (b4 - b5) = (dom b4) /\ (dom b5) & integral' b3,(b4 | (dom (b4 - b5))) = (integral' b3,(b4 - b5)) + (integral' b3,(b5 | (dom (b4 - b5)))) )
proof end;

theorem Th76: :: MESFUNC5:76
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & b4 is nonnegative & b5 is nonnegative & ( for b6 being set st b6 in dom (b4 - b5) holds
b5 . b6 <= b4 . b6 ) holds
integral' b3,(b5 | (dom (b4 - b5))) <= integral' b3,(b4 | (dom (b4 - b5)))
proof end;

theorem Th77: :: MESFUNC5:77
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being R_eal st 0 <= b5 & b4 is_simple_func_in b2 & ( for b6 being set st b6 in dom b4 holds
b4 . b6 = b5 ) holds
integral' b3,b4 = b5 * (b3 . (dom b4))
proof end;

theorem Th78: :: MESFUNC5:78
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b4 is nonnegative holds
integral' b3,(b4 | (eq_dom b4,(R_EAL 0))) = 0
proof end;

theorem Th79: :: MESFUNC5:79
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5 being PartFunc of b1, ExtREAL st b5 is_simple_func_in b2 & b3 . b4 = 0 & b5 is nonnegative holds
integral' b3,(b5 | b4) = 0
proof end;

theorem Th80: :: MESFUNC5:80
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Functional_Sequence of b1, ExtREAL
for b6 being ExtREAL_sequence st b4 is_simple_func_in b2 & ( for b7 being set st b7 in dom b4 holds
0 < b4 . b7 ) & ( for b7 being Nat holds b5 . b7 is_simple_func_in b2 ) & ( for b7 being Nat holds dom (b5 . b7) = dom b4 ) & ( for b7 being Nat holds b5 . b7 is nonnegative ) & ( for b7, b8 being Nat st b7 <= b8 holds
for b9 being Element of b1 st b9 in dom b4 holds
(b5 . b7) . b9 <= (b5 . b8) . b9 ) & ( for b7 being Element of b1 st b7 in dom b4 holds
( b5 # b7 is convergent & b4 . b7 <= lim (b5 # b7) ) ) & ( for b7 being Nat holds b6 . b7 = integral' b3,(b5 . b7) ) holds
( b6 is convergent & integral' b3,b4 <= lim b6 )
proof end;

theorem Th81: :: MESFUNC5:81
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Functional_Sequence of b1, ExtREAL st b4 is_simple_func_in b2 & b4 is nonnegative & ( for b6 being Nat holds b5 . b6 is_simple_func_in b2 ) & ( for b6 being Nat holds dom (b5 . b6) = dom b4 ) & ( for b6 being Nat holds b5 . b6 is nonnegative ) & ( for b6, b7 being Nat st b6 <= b7 holds
for b8 being Element of b1 st b8 in dom b4 holds
(b5 . b6) . b8 <= (b5 . b7) . b8 ) & ( for b6 being Element of b1 st b6 in dom b4 holds
( b5 # b6 is convergent & b4 . b6 <= lim (b5 # b6) ) ) holds
ex b6 being ExtREAL_sequence st
( ( for b7 being Nat holds b6 . b7 = integral' b3,(b5 . b7) ) & b6 is convergent & sup (rng b6) = lim b6 & integral' b3,b4 <= lim b6 )
proof end;

theorem Th82: :: MESFUNC5:82
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being Functional_Sequence of b1, ExtREAL
for b7, b8 being ExtREAL_sequence st ( for b9 being Nat holds
( b5 . b9 is_simple_func_in b2 & dom (b5 . b9) = b4 ) ) & ( for b9 being Nat holds b5 . b9 is nonnegative ) & ( for b9, b10 being Nat st b9 <= b10 holds
for b11 being Element of b1 st b11 in b4 holds
(b5 . b9) . b11 <= (b5 . b10) . b11 ) & ( for b9 being Nat holds
( b6 . b9 is_simple_func_in b2 & dom (b6 . b9) = b4 ) ) & ( for b9 being Nat holds b6 . b9 is nonnegative ) & ( for b9, b10 being Nat st b9 <= b10 holds
for b11 being Element of b1 st b11 in b4 holds
(b6 . b9) . b11 <= (b6 . b10) . b11 ) & ( for b9 being Element of b1 st b9 in b4 holds
( b5 # b9 is convergent & b6 # b9 is convergent & lim (b5 # b9) = lim (b6 # b9) ) ) & ( for b9 being Nat holds
( b7 . b9 = integral' b3,(b5 . b9) & b8 . b9 = integral' b3,(b6 . b9) ) ) holds
( b7 is convergent & b8 is convergent & lim b7 = lim b8 )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
assume that
E101: ex b1 being Element of c2 st
( b1 = dom c4 & c4 is_measurable_on b1 ) and
E102: c4 is nonnegative ;
func integral+ c3,c4 -> Element of ExtREAL means :Def15: :: MESFUNC5:def 15
ex b1 being Functional_Sequence of a1, ExtREAL ex b2 being ExtREAL_sequence st
( ( for b3 being Nat holds
( b1 . b3 is_simple_func_in a2 & dom (b1 . b3) = dom a4 ) ) & ( for b3 being Nat holds b1 . b3 is nonnegative ) & ( for b3, b4 being Nat st b3 <= b4 holds
for b5 being Element of a1 st b5 in dom a4 holds
(b1 . b3) . b5 <= (b1 . b4) . b5 ) & ( for b3 being Element of a1 st b3 in dom a4 holds
( b1 # b3 is convergent & lim (b1 # b3) = a4 . b3 ) ) & ( for b3 being Nat holds b2 . b3 = integral' a3,(b1 . b3) ) & b2 is convergent & a5 = lim b2 );
existence
ex b1 being Element of ExtREAL ex b2 being Functional_Sequence of c1, ExtREAL ex b3 being ExtREAL_sequence st
( ( for b4 being Nat holds
( b2 . b4 is_simple_func_in c2 & dom (b2 . b4) = dom c4 ) ) & ( for b4 being Nat holds b2 . b4 is nonnegative ) & ( for b4, b5 being Nat st b4 <= b5 holds
for b6 being Element of c1 st b6 in dom c4 holds
(b2 . b4) . b6 <= (b2 . b5) . b6 ) & ( for b4 being Element of c1 st b4 in dom c4 holds
( b2 # b4 is convergent & lim (b2 # b4) = c4 . b4 ) ) & ( for b4 being Nat holds b3 . b4 = integral' c3,(b2 . b4) ) & b3 is convergent & b1 = lim b3 )
proof end;
uniqueness
for b1, b2 being Element of ExtREAL st ex b3 being Functional_Sequence of c1, ExtREAL ex b4 being ExtREAL_sequence st
( ( for b5 being Nat holds
( b3 . b5 is_simple_func_in c2 & dom (b3 . b5) = dom c4 ) ) & ( for b5 being Nat holds b3 . b5 is nonnegative ) & ( for b5, b6 being Nat st b5 <= b6 holds
for b7 being Element of c1 st b7 in dom c4 holds
(b3 . b5) . b7 <= (b3 . b6) . b7 ) & ( for b5 being Element of c1 st b5 in dom c4 holds
( b3 # b5 is convergent & lim (b3 # b5) = c4 . b5 ) ) & ( for b5 being Nat holds b4 . b5 = integral' c3,(b3 . b5) ) & b4 is convergent & b1 = lim b4 ) & ex b3 being Functional_Sequence of c1, ExtREAL ex b4 being ExtREAL_sequence st
( ( for b5 being Nat holds
( b3 . b5 is_simple_func_in c2 & dom (b3 . b5) = dom c4 ) ) & ( for b5 being Nat holds b3 . b5 is nonnegative ) & ( for b5, b6 being Nat st b5 <= b6 holds
for b7 being Element of c1 st b7 in dom c4 holds
(b3 . b5) . b7 <= (b3 . b6) . b7 ) & ( for b5 being Element of c1 st b5 in dom c4 holds
( b3 # b5 is convergent & lim (b3 # b5) = c4 . b5 ) ) & ( for b5 being Nat holds b4 . b5 = integral' c3,(b3 . b5) ) & b4 is convergent & b2 = lim b4 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def15 defines integral+ MESFUNC5:def 15 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative holds
for b5 being Element of ExtREAL holds
( b5 = integral+ b3,b4 iff ex b6 being Functional_Sequence of b1, ExtREAL ex b7 being ExtREAL_sequence st
( ( for b8 being Nat holds
( b6 . b8 is_simple_func_in b2 & dom (b6 . b8) = dom b4 ) ) & ( for b8 being Nat holds b6 . b8 is nonnegative ) & ( for b8, b9 being Nat st b8 <= b9 holds
for b10 being Element of b1 st b10 in dom b4 holds
(b6 . b8) . b10 <= (b6 . b9) . b10 ) & ( for b8 being Element of b1 st b8 in dom b4 holds
( b6 # b8 is convergent & lim (b6 # b8) = b4 . b8 ) ) & ( for b8 being Nat holds b7 . b8 = integral' b3,(b6 . b8) ) & b7 is convergent & b5 = lim b7 ) );

theorem Th83: :: MESFUNC5:83
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b4 is nonnegative holds
integral+ b3,b4 = integral' b3,b4
proof end;

Lemma103: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b6 = dom b5 & b4 is_measurable_on b6 & b5 is_measurable_on b6 ) & b4 is nonnegative & b5 is nonnegative holds
integral+ b3,(b4 + b5) = (integral+ b3,b4) + (integral+ b3,b5)
proof end;

theorem Th84: :: MESFUNC5:84
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & ex b6 being Element of b2 st
( b6 = dom b5 & b5 is_measurable_on b6 ) & b4 is nonnegative & b5 is nonnegative holds
ex b6 being Element of b2 st
( b6 = dom (b4 + b5) & integral+ b3,(b4 + b5) = (integral+ b3,(b4 | b6)) + (integral+ b3,(b5 | b6)) )
proof end;

theorem Th85: :: MESFUNC5:85
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative holds
0 <= integral+ b3,b4
proof end;

theorem Th86: :: MESFUNC5:86
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative holds
0 <= integral+ b3,(b4 | b5)
proof end;

theorem Th87: :: MESFUNC5:87
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 misses b6 holds
integral+ b3,(b4 | (b5 \/ b6)) = (integral+ b3,(b4 | b5)) + (integral+ b3,(b4 | b6))
proof end;

theorem Th88: :: MESFUNC5:88
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative & b3 . b5 = 0 holds
integral+ b3,(b4 | b5) = 0
proof end;

theorem Th89: :: MESFUNC5:89
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 c= b6 holds
integral+ b3,(b4 | b5) <= integral+ b3,(b4 | b6)
proof end;

theorem Th90: :: MESFUNC5:90
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st b4 is nonnegative & b5 = dom b4 & b4 is_measurable_on b5 & b3 . b6 = 0 holds
integral+ b3,(b4 | (b5 \ b6)) = integral+ b3,b4
proof end;

theorem Th91: :: MESFUNC5:91
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b6 = dom b5 & b4 is_measurable_on b6 & b5 is_measurable_on b6 ) & b4 is nonnegative & b5 is nonnegative & ( for b6 being Element of b1 st b6 in dom b5 holds
b5 . b6 <= b4 . b6 ) holds
integral+ b3,b5 <= integral+ b3,b4
proof end;

theorem Th92: :: MESFUNC5:92
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st 0 <= b5 & ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative holds
integral+ b3,(b5 (#) b4) = (R_EAL b5) * (integral+ b3,b4)
proof end;

theorem Th93: :: MESFUNC5:93
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & ( for b5 being Element of b1 st b5 in dom b4 holds
0 = b4 . b5 ) holds
integral+ b3,b4 = 0
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
func Integral c3,c4 -> Element of ExtREAL equals :: MESFUNC5:def 16
(integral+ a3,(max+ a4)) - (integral+ a3,(max- a4));
coherence
(integral+ c3,(max+ c4)) - (integral+ c3,(max- c4)) is Element of ExtREAL
;
end;

:: deftheorem Def16 defines Integral MESFUNC5:def 16 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds Integral b3,b4 = (integral+ b3,(max+ b4)) - (integral+ b3,(max- b4));

theorem Th94: :: MESFUNC5:94
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative holds
Integral b3,b4 = integral+ b3,b4
proof end;

theorem Th95: :: MESFUNC5:95
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_simple_func_in b2 & b4 is nonnegative holds
( Integral b3,b4 = integral+ b3,b4 & Integral b3,b4 = integral' b3,b4 )
proof end;

theorem Th96: :: MESFUNC5:96
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative holds
0 <= Integral b3,b4
proof end;

theorem Th97: :: MESFUNC5:97
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 misses b6 holds
Integral b3,(b4 | (b5 \/ b6)) = (Integral b3,(b4 | b5)) + (Integral b3,(b4 | b6))
proof end;

theorem Th98: :: MESFUNC5:98
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative holds
0 <= Integral b3,(b4 | b5)
proof end;

theorem Th99: :: MESFUNC5:99
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 c= b6 holds
Integral b3,(b4 | b5) <= Integral b3,(b4 | b6)
proof end;

theorem Th100: :: MESFUNC5:100
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b3 . b5 = 0 holds
Integral b3,(b4 | b5) = 0
proof end;

theorem Th101: :: MESFUNC5:101
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st b5 = dom b4 & b4 is_measurable_on b5 & b3 . b6 = 0 holds
Integral b3,(b4 | (b5 \ b6)) = Integral b3,b4
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
pred c4 is_integrable_on c3 means :Def17: :: MESFUNC5:def 17
( ex b1 being Element of a2 st
( b1 = dom a4 & a4 is_measurable_on b1 ) & integral+ a3,(max+ a4) < +infty & integral+ a3,(max- a4) < +infty );
end;

:: deftheorem Def17 defines is_integrable_on MESFUNC5:def 17 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 iff ( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & integral+ b3,(max+ b4) < +infty & integral+ b3,(max- b4) < +infty ) );

theorem Th102: :: MESFUNC5:102
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 holds
( 0 <= integral+ b3,(max+ b4) & 0 <= integral+ b3,(max- b4) & -infty < Integral b3,b4 & Integral b3,b4 < +infty )
proof end;

theorem Th103: :: MESFUNC5:103
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 st b4 is_integrable_on b3 holds
( integral+ b3,(max+ (b4 | b5)) <= integral+ b3,(max+ b4) & integral+ b3,(max- (b4 | b5)) <= integral+ b3,(max- b4) & b4 | b5 is_integrable_on b3 )
proof end;

theorem Th104: :: MESFUNC5:104
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st b4 is_integrable_on b3 & b5 misses b6 holds
Integral b3,(b4 | (b5 \/ b6)) = (Integral b3,(b4 | b5)) + (Integral b3,(b4 | b6))
proof end;

theorem Th105: :: MESFUNC5:105
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 st b4 is_integrable_on b3 & b6 = (dom b4) \ b5 holds
( b4 | b5 is_integrable_on b3 & Integral b3,b4 = (Integral b3,(b4 | b5)) + (Integral b3,(b4 | b6)) )
proof end;

theorem Th106: :: MESFUNC5:106
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) holds
( b4 is_integrable_on b3 iff |.b4.| is_integrable_on b3 )
proof end;

theorem Th107: :: MESFUNC5:107
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 holds
|.(Integral b3,b4).| <= Integral b3,|.b4.|
proof end;

theorem Th108: :: MESFUNC5:108
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & dom b4 = dom b5 & b5 is_integrable_on b3 & ( for b6 being Element of b1 st b6 in dom b4 holds
|.(b4 . b6).| <= b5 . b6 ) holds
( b4 is_integrable_on b3 & Integral b3,|.b4.| <= Integral b3,b5 )
proof end;

theorem Th109: :: MESFUNC5:109
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st dom b4 in b2 & 0 <= b5 & dom b4 <> {} & ( for b6 being set st b6 in dom b4 holds
b4 . b6 = b5 ) holds
integral b1,b2,b3,b4 = (R_EAL b5) * (b3 . (dom b4))
proof end;

theorem Th110: :: MESFUNC5:110
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st dom b4 in b2 & 0 <= b5 & ( for b6 being set st b6 in dom b4 holds
b4 . b6 = b5 ) holds
integral' b3,b4 = (R_EAL b5) * (b3 . (dom b4))
proof end;

theorem Th111: :: MESFUNC5:111
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 holds
( b4 " {+infty } in b2 & b4 " {-infty } in b2 & b3 . (b4 " {+infty }) = 0 & b3 . (b4 " {-infty }) = 0 & (b4 " {+infty }) \/ (b4 " {-infty }) in b2 & b3 . ((b4 " {+infty }) \/ (b4 " {-infty })) = 0 )
proof end;

theorem Th112: :: MESFUNC5:112
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 & b5 is_integrable_on b3 & b4 is nonnegative & b5 is nonnegative holds
b4 + b5 is_integrable_on b3
proof end;

theorem Th113: :: MESFUNC5:113
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 & b5 is_integrable_on b3 holds
dom (b4 + b5) in b2
proof end;

Lemma127: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st ex b6 being Element of b2 st
( dom (b4 + b5) = b6 & b4 + b5 is_measurable_on b6 ) & b4 is_integrable_on b3 & b5 is_integrable_on b3 holds
b4 + b5 is_integrable_on b3
proof end;

theorem Th114: :: MESFUNC5:114
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 & b5 is_integrable_on b3 holds
b4 + b5 is_integrable_on b3
proof end;

Lemma129: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 & b5 is_integrable_on b3 & dom b4 = dom b5 holds
ex b6, b7, b8 being Element of b2 st
( b6 c= dom b4 & b7 c= dom b4 & b6 = (dom b4) \ b7 & b4 | b6 is_finite & b6 = dom (b4 | b6) & b4 | b6 is_measurable_on b6 & b4 | b6 is_integrable_on b3 & Integral b3,b4 = Integral b3,(b4 | b6) & b6 c= dom b5 & b7 c= dom b5 & b6 = (dom b5) \ b7 & b5 | b6 is_finite & b6 = dom (b5 | b6) & b5 | b6 is_measurable_on b6 & b5 | b6 is_integrable_on b3 & Integral b3,b5 = Integral b3,(b5 | b6) & b6 c= dom (b4 + b5) & b8 c= dom (b4 + b5) & b6 = (dom (b4 + b5)) \ b8 & b3 . b7 = 0 & b3 . b8 = 0 & b6 = dom ((b4 + b5) | b6) & (b4 + b5) | b6 is_measurable_on b6 & (b4 + b5) | b6 is_integrable_on b3 & (b4 + b5) | b6 = (b4 | b6) + (b5 | b6) & Integral b3,((b4 + b5) | b6) = (Integral b3,(b4 | b6)) + (Integral b3,(b5 | b6)) )
proof end;

Lemma130: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 & b5 is_integrable_on b3 & dom b4 = dom b5 holds
( b4 + b5 is_integrable_on b3 & Integral b3,(b4 + b5) = (Integral b3,b4) + (Integral b3,b5) )
proof end;

theorem Th115: :: MESFUNC5:115
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL st b4 is_integrable_on b3 & b5 is_integrable_on b3 holds
ex b6 being Element of b2 st
( b6 = (dom b4) /\ (dom b5) & Integral b3,(b4 + b5) = (Integral b3,(b4 | b6)) + (Integral b3,(b5 | b6)) )
proof end;

theorem Th116: :: MESFUNC5:116
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real st b4 is_integrable_on b3 holds
( b5 (#) b4 is_integrable_on b3 & Integral b3,(b5 (#) b4) = (R_EAL b5) * (Integral b3,b4) )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
let c5 be Element of c2;
func Integral_on c3,c5,c4 -> Element of ExtREAL equals :: MESFUNC5:def 18
Integral a3,(a4 | a5);
coherence
Integral c3,(c4 | c5) is Element of ExtREAL
;
end;

:: deftheorem Def18 defines Integral_on MESFUNC5:def 18 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds Integral_on b3,b5,b4 = Integral b3,(b4 | b5);

theorem Th117: :: MESFUNC5:117
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL
for b6 being Element of b2 st b4 is_integrable_on b3 & b5 is_integrable_on b3 & b6 c= dom (b4 + b5) holds
( b4 + b5 is_integrable_on b3 & Integral_on b3,b6,(b4 + b5) = (Integral_on b3,b6,b4) + (Integral_on b3,b6,b5) )
proof end;

theorem Th118: :: MESFUNC5:118
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real
for b6 being Element of b2 st b4 is_integrable_on b3 & b4 is_measurable_on b6 holds
( b4 | b6 is_integrable_on b3 & Integral_on b3,b6,(b5 (#) b4) = (R_EAL b5) * (Integral_on b3,b6,b4) )
proof end;