:: MESFUNC3 semantic presentation
theorem Th1: :: MESFUNC3:1
theorem Th2: :: MESFUNC3:2
theorem Th3: :: MESFUNC3:3
theorem Th4: :: MESFUNC3:4
theorem Th5: :: MESFUNC3:5
theorem Th6: :: MESFUNC3:6
theorem Th7: :: MESFUNC3:7
theorem Th8: :: MESFUNC3:8
theorem Th9: :: MESFUNC3:9
theorem Th10: :: MESFUNC3:10
theorem Th11: :: MESFUNC3:11
:: deftheorem Def1 defines are_Re-presentation_of MESFUNC3:def 1 :
theorem Th12: :: MESFUNC3:12
theorem Th13: :: MESFUNC3:13
Lemma14:
for b1 being FinSequence of ExtREAL
for b2, b3 being Element of ExtREAL holds
( b3 = len b1 & ( for b4 being Nat holds
( b4 in dom b1 implies b1 . b4 = b2 ) ) implies Sum b1 = b3 * b2 )
Lemma15:
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL holds
not ( b3 is_simple_func_in b2 & ( for b4 being set holds
( b4 in dom b3 implies 0. <= b3 . b4 ) ) & ( for b4 being set holds
not ( b4 in dom b3 & not 0. <> b3 . b4 ) ) & ( for b4 being Finite_Sep_Sequence of b2
for b5 being FinSequence of ExtREAL holds
not ( b4,b5 are_Re-presentation_of b3 & b5 . 1 = 0. & ( for b6 being Nat holds
( 2 <= b6 & b6 in dom b5 implies ( 0. < b5 . b6 & b5 . b6 < +infty ) ) ) ) ) )
Lemma16:
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL holds
not ( b3 is_simple_func_in b2 & ( for b4 being set holds
( b4 in dom b3 implies 0. <= b3 . b4 ) ) & ex b4 being set st
( b4 in dom b3 & 0. = b3 . b4 ) & ( for b4 being Finite_Sep_Sequence of b2
for b5 being FinSequence of ExtREAL holds
not ( b4,b5 are_Re-presentation_of b3 & b5 . 1 = 0. & ( for b6 being Nat holds
( 2 <= b6 & b6 in dom b5 implies ( 0. < b5 . b6 & b5 . b6 < +infty ) ) ) ) ) )
theorem Th14: :: MESFUNC3:14
theorem Th15: :: MESFUNC3:15
theorem Th16: :: MESFUNC3:16
theorem Th17: :: MESFUNC3:17
definition
let c
1 be non
empty set ;
let c
2 be
SigmaField of c
1;
let c
3 be
sigma_Measure of c
2;
let c
4 be
PartFunc of c
1,
ExtREAL ;
assume E19:
( c
4 is_simple_func_in c
2 &
dom c
4 <> {} & ( for b
1 being
set holds
( b
1 in dom c
4 implies
0. <= c
4 . b
1 ) ) )
;
func integral c
1,c
2,c
3,c
4 -> Element of
ExtREAL means :: MESFUNC3:def 2
ex b
1 being
Finite_Sep_Sequence of a
2ex b
2, b
3 being
FinSequence of
ExtREAL st
( b
1,b
2 are_Re-presentation_of a
4 & b
2 . 1
= 0. & ( for b
4 being
Nat holds
( 2
<= b
4 & b
4 in dom b
2 implies (
0. < b
2 . b
4 & b
2 . b
4 < +infty ) ) ) &
dom b
3 = dom b
1 & ( for b
4 being
Nat holds
( b
4 in dom b
3 implies b
3 . b
4 = (b2 . b4) * ((a3 * b1) . b4) ) ) & a
5 = Sum b
3 );
existence
ex b1 being Element of ExtREAL ex b2 being Finite_Sep_Sequence of c2ex b3, b4 being FinSequence of ExtREAL st
( b2,b3 are_Re-presentation_of c4 & b3 . 1 = 0. & ( for b5 being Nat holds
( 2 <= b5 & b5 in dom b3 implies ( 0. < b3 . b5 & b3 . b5 < +infty ) ) ) & dom b4 = dom b2 & ( for b5 being Nat holds
( b5 in dom b4 implies b4 . b5 = (b3 . b5) * ((c3 * b2) . b5) ) ) & b1 = Sum b4 )
uniqueness
for b1, b2 being Element of ExtREAL holds
( ex b3 being Finite_Sep_Sequence of c2ex b4, b5 being FinSequence of ExtREAL st
( b3,b4 are_Re-presentation_of c4 & b4 . 1 = 0. & ( for b6 being Nat holds
( 2 <= b6 & b6 in dom b4 implies ( 0. < b4 . b6 & b4 . b6 < +infty ) ) ) & dom b5 = dom b3 & ( for b6 being Nat holds
( b6 in dom b5 implies b5 . b6 = (b4 . b6) * ((c3 * b3) . b6) ) ) & b1 = Sum b5 ) & ex b3 being Finite_Sep_Sequence of c2ex b4, b5 being FinSequence of ExtREAL st
( b3,b4 are_Re-presentation_of c4 & b4 . 1 = 0. & ( for b6 being Nat holds
( 2 <= b6 & b6 in dom b4 implies ( 0. < b4 . b6 & b4 . b6 < +infty ) ) ) & dom b5 = dom b3 & ( for b6 being Nat holds
( b6 in dom b5 implies b5 . b6 = (b4 . b6) * ((c3 * b3) . b6) ) ) & b2 = Sum b5 ) implies b1 = b2 )
end;
:: deftheorem Def2 defines integral MESFUNC3:def 2 :
theorem Th18: :: MESFUNC3:18