:: MESFUNC5 semantic presentation
theorem Th1: :: MESFUNC5:1
theorem Th2: :: MESFUNC5:2
theorem Th3: :: MESFUNC5:3
theorem Th4: :: MESFUNC5:4
for b
1, b
2 being
R_eal holds
( ( for b
3 being
real number holds
not ( 0
< b
3 & not b
1 < b
2 + (R_EAL b3) ) ) implies b
1 <= b
2 )
theorem Th5: :: MESFUNC5:5
theorem Th6: :: MESFUNC5:6
theorem Th7: :: MESFUNC5:7
theorem Th8: :: MESFUNC5:8
for b
1 being
Natfor b
2 being
R_eal holds
not ( 0
<= b
2 & b
2 < b
1 & ( for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 <= (2 |^ b1) * b
1 &
(b3 - 1) / (2 |^ b1) <= b
2 & b
2 < b
3 / (2 |^ b1) ) ) )
theorem Th9: :: MESFUNC5:9
for b
1, b
2 being
Natfor b
3 being
R_eal holds
( 1
<= b
2 & b
2 <= (2 |^ b1) * b
1 & b
1 <= b
3 &
(b2 - 1) / (2 |^ b1) <= b
3 implies b
2 / (2 |^ b1) <= b
3 )
theorem Th10: :: MESFUNC5:10
for b
1, b
2, b
3, b
4 being
R_eal holds
not (
-infty < b
3 & b
1 < b
2 & b
3 < b
4 & not b
1 + b
3 < b
2 + b
4 )
theorem Th11: :: MESFUNC5:11
for b
1, b
2, b
3 being
R_eal holds
( 0
<= b
3 implies ( b
3 * (max b1,b2) = max (b3 * b1),
(b3 * b2) & b
3 * (min b1,b2) = min (b3 * b1),
(b3 * b2) ) )
theorem Th12: :: MESFUNC5:12
for b
1, b
2, b
3 being
R_eal holds
( b
3 <= 0 implies ( b
3 * (min b1,b2) = max (b3 * b1),
(b3 * b2) & b
3 * (max b1,b2) = min (b3 * b1),
(b3 * b2) ) )
theorem Th13: :: MESFUNC5:13
for b
1, b
2, b
3 being
R_eal holds
( 0
<= b
1 & 0
<= b
3 & b
3 + b
1 <= b
2 implies b
3 <= b
2 )
:: deftheorem Def1 defines nonpositive MESFUNC5:def 1 :
:: deftheorem Def2 defines nonpositive MESFUNC5:def 2 :
theorem Th14: :: MESFUNC5:14
theorem Th15: :: MESFUNC5:15
:: deftheorem Def3 defines without-infty MESFUNC5:def 3 :
:: deftheorem Def4 defines without+infty MESFUNC5:def 4 :
:: deftheorem Def5 defines without-infty MESFUNC5:def 5 :
:: deftheorem Def6 defines without+infty MESFUNC5:def 6 :
theorem Th16: :: MESFUNC5:16
theorem Th17: :: MESFUNC5:17
theorem Th18: :: MESFUNC5:18
theorem Th19: :: MESFUNC5:19
theorem Th20: :: MESFUNC5:20
theorem Th21: :: MESFUNC5:21
theorem Th22: :: MESFUNC5:22
theorem Th23: :: MESFUNC5:23
theorem Th24: :: MESFUNC5:24
:: deftheorem Def7 defines R_EAL MESFUNC5:def 7 :
theorem Th25: :: MESFUNC5:25
theorem Th26: :: MESFUNC5:26
theorem Th27: :: MESFUNC5:27
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 )
theorem Th28: :: MESFUNC5:28
theorem Th29: :: MESFUNC5:29
theorem Th30: :: MESFUNC5:30
theorem Th31: :: MESFUNC5:31
theorem Th32: :: MESFUNC5:32
theorem Th33: :: MESFUNC5:33
theorem Th34: :: MESFUNC5:34
theorem Th35: :: MESFUNC5:35
theorem Th36: :: MESFUNC5:36
theorem Th37: :: MESFUNC5:37
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 holds
( b3 is nonnegative & b4 is nonnegative & b3 is_measurable_on b5 & b4 is_measurable_on b5 implies ( dom (b3 + b4) = (dom b3) /\ (dom b4) & b3 + b4 is_measurable_on b5 ) )
theorem Th38: :: MESFUNC5:38
theorem Th39: :: MESFUNC5:39
theorem Th40: :: MESFUNC5:40
theorem Th41: :: MESFUNC5:41
theorem Th42: :: MESFUNC5:42
theorem Th43: :: MESFUNC5:43
Lemma48:
for b1 being set
for b2 being FinSequence holds
( ( for b3 being Nat holds
( b3 in dom b2 implies b2 . b3 in b1 ) ) implies b2 is FinSequence of b1 )
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 holds
( b4 is_simple_func_in b2 & dom b4 <> {} & b5 is_simple_func_in b2 & dom b5 = dom b4 implies ( b4 + b5 is_simple_func_in b2 & dom (b4 + b5) <> {} ) )
theorem Th44: :: MESFUNC5:44
theorem Th45: :: MESFUNC5:45
theorem Th46: :: MESFUNC5:46
theorem Th47: :: MESFUNC5:47
theorem Th48: :: MESFUNC5:48
theorem Th49: :: MESFUNC5:49
theorem Th50: :: MESFUNC5:50
theorem Th51: :: MESFUNC5:51
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 holds
( dom b4 in b2 & ( for b6 being set holds
( b6 in dom b4 implies b4 . b6 = b5 ) ) implies b4 is_simple_func_in b2 )
theorem Th52: :: MESFUNC5:52
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)
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 )
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 holds
not ( 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 & ( for b6 being Element of b2 holds
not ( b6 = dom (b4 + b5) & b4 + b5 is_measurable_on b6 ) ) )
theorem Th53: :: MESFUNC5:53
theorem Th54: :: MESFUNC5:54
theorem Th55: :: MESFUNC5:55
:: deftheorem Def8 defines convergent_to_finite_number MESFUNC5:def 8 :
:: deftheorem Def9 defines convergent_to_+infty MESFUNC5:def 9 :
:: deftheorem Def10 defines convergent_to_-infty MESFUNC5:def 10 :
theorem Th56: :: MESFUNC5:56
theorem Th57: :: MESFUNC5:57
:: deftheorem Def11 defines convergent MESFUNC5:def 11 :
:: deftheorem Def12 defines lim MESFUNC5:def 12 :
theorem Th58: :: MESFUNC5:58
theorem Th59: :: MESFUNC5:59
theorem Th60: :: MESFUNC5:60
theorem Th61: :: MESFUNC5:61
theorem Th62: :: MESFUNC5:62
theorem Th63: :: MESFUNC5:63
theorem Th64: :: MESFUNC5:64
theorem Th65: :: MESFUNC5:65
theorem Th66: :: MESFUNC5:66
Lemma81:
for b1 being ExtREAL_sequence holds
not ( b1 is without-infty & not sup (rng b1) in REAL & not sup (rng b1) = +infty )
theorem Th67: :: MESFUNC5:67
theorem Th68: :: MESFUNC5:68
theorem Th69: :: MESFUNC5:69
:: deftheorem Def13 defines # MESFUNC5:def 13 :
theorem Th70: :: MESFUNC5:70
:: deftheorem Def14 defines integral' MESFUNC5:def 14 :
theorem Th71: :: MESFUNC5:71
theorem Th72: :: MESFUNC5:72
theorem Th73: :: MESFUNC5:73
theorem Th74: :: MESFUNC5:74
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 holds
( 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 holds
( b6 in dom b4 implies b5 . b6 <= b4 . b6 ) ) implies ( 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) ) )
theorem Th75: :: MESFUNC5:75
theorem Th76: :: MESFUNC5:76
theorem Th77: :: MESFUNC5:77
theorem Th78: :: MESFUNC5:78
theorem Th79: :: MESFUNC5:79
theorem Th80: :: MESFUNC5:80
theorem Th81: :: MESFUNC5:81
theorem Th82: :: MESFUNC5:82
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 that E101:
ex b
1 being
Element of c
2 st
( b
1 = dom c
4 & c
4 is_measurable_on b
1 )
and E102:
c
4 is
nonnegative
;
func integral+ c
3,c
4 -> Element of
ExtREAL means :
Def15:
:: MESFUNC5:def 15
ex b
1 being
Functional_Sequence of a
1,
ExtREAL ex b
2 being
ExtREAL_sequence st
( ( for b
3 being
Nat holds
( b
1 . b
3 is_simple_func_in a
2 &
dom (b1 . b3) = dom a
4 ) ) & ( for b
3 being
Nat holds b
1 . b
3 is
nonnegative ) & ( for b
3, b
4 being
Nat holds
( b
3 <= b
4 implies for b
5 being
Element of a
1 holds
( b
5 in dom a
4 implies
(b1 . b3) . b
5 <= (b1 . b4) . b
5 ) ) ) & ( for b
3 being
Element of a
1 holds
( b
3 in dom a
4 implies ( b
1 # b
3 is
convergent &
lim (b1 # b3) = a
4 . b
3 ) ) ) & ( for b
3 being
Nat holds b
2 . b
3 = integral' a
3,
(b1 . b3) ) & b
2 is
convergent & a
5 = lim b
2 );
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 holds
( b4 <= b5 implies for b6 being Element of c1 holds
( b6 in dom c4 implies (b2 . b4) . b6 <= (b2 . b5) . b6 ) ) ) & ( for b4 being Element of c1 holds
( b4 in dom c4 implies ( 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 )
uniqueness
for b1, b2 being Element of ExtREAL holds
( 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 holds
( b5 <= b6 implies for b7 being Element of c1 holds
( b7 in dom c4 implies (b3 . b5) . b7 <= (b3 . b6) . b7 ) ) ) & ( for b5 being Element of c1 holds
( b5 in dom c4 implies ( 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 holds
( b5 <= b6 implies for b7 being Element of c1 holds
( b7 in dom c4 implies (b3 . b5) . b7 <= (b3 . b6) . b7 ) ) ) & ( for b5 being Element of c1 holds
( b5 in dom c4 implies ( 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 ) implies b1 = b2 )
end;
:: deftheorem Def15 defines integral+ MESFUNC5:def 15 :
theorem Th83: :: MESFUNC5:83
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 holds
( 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 implies integral+ b3,(b4 + b5) = (integral+ b3,b4) + (integral+ b3,b5) )
theorem Th84: :: MESFUNC5:84
theorem Th85: :: MESFUNC5:85
theorem Th86: :: MESFUNC5:86
theorem Th87: :: MESFUNC5:87
theorem Th88: :: MESFUNC5:88
theorem Th89: :: MESFUNC5:89
theorem Th90: :: MESFUNC5:90
theorem Th91: :: MESFUNC5:91
theorem Th92: :: MESFUNC5:92
theorem Th93: :: MESFUNC5:93
:: deftheorem Def16 defines Integral MESFUNC5:def 16 :
theorem Th94: :: MESFUNC5:94
theorem Th95: :: MESFUNC5:95
theorem Th96: :: MESFUNC5:96
theorem Th97: :: MESFUNC5:97
theorem Th98: :: MESFUNC5:98
theorem Th99: :: MESFUNC5:99
theorem Th100: :: MESFUNC5:100
theorem Th101: :: MESFUNC5:101
:: deftheorem Def17 defines is_integrable_on MESFUNC5:def 17 :
theorem Th102: :: MESFUNC5:102
theorem Th103: :: MESFUNC5:103
theorem Th104: :: MESFUNC5:104
theorem Th105: :: MESFUNC5:105
theorem Th106: :: MESFUNC5:106
theorem Th107: :: MESFUNC5:107
theorem Th108: :: MESFUNC5:108
theorem Th109: :: MESFUNC5:109
theorem Th110: :: MESFUNC5:110
theorem Th111: :: MESFUNC5:111
theorem Th112: :: MESFUNC5:112
theorem Th113: :: MESFUNC5:113
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 holds
( 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 implies b4 + b5 is_integrable_on b3 )
theorem Th114: :: MESFUNC5:114
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 holds
not ( b4 is_integrable_on b3 & b5 is_integrable_on b3 & dom b4 = dom b5 & ( for b6, b7, b8 being Element of b2 holds
not ( 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)) ) ) )
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 holds
( b4 is_integrable_on b3 & b5 is_integrable_on b3 & dom b4 = dom b5 implies ( b4 + b5 is_integrable_on b3 & Integral b3,(b4 + b5) = (Integral b3,b4) + (Integral b3,b5) ) )
theorem Th115: :: MESFUNC5:115
theorem Th116: :: MESFUNC5:116
:: deftheorem Def18 defines Integral_on MESFUNC5:def 18 :
theorem Th117: :: MESFUNC5:117
theorem Th118: :: MESFUNC5:118