Article MESFUNC3, MML version 4.99.1005
:: MESFUNC3:th 1
theorem
for b1, b2 being natural set
for b3 being Function-like quasi_total Relation of [:Seg b1,Seg b2:],REAL
for b4, b5 being FinSequence of REAL
st dom b4 = Seg b1 &
(for b6 being natural set
st b6 in dom b4
holds ex b7 being FinSequence of REAL st
dom b7 = Seg b2 &
b4 . b6 = Sum b7 &
(for b8 being natural set
st b8 in dom b7
holds b7 . b8 = b3 . [b6,b8])) &
dom b5 = Seg b2 &
(for b6 being natural set
st b6 in dom b5
holds ex b7 being FinSequence of REAL st
dom b7 = Seg b1 &
b5 . b6 = Sum b7 &
(for b8 being natural set
st b8 in dom b7
holds b7 . b8 = b3 . [b8,b6]))
holds Sum b4 = Sum b5;
:: MESFUNC3:th 2
theorem
for b1 being FinSequence of ExtREAL
for b2 being FinSequence of REAL
st b1 = b2
holds Sum b1 = Sum b2;
:: MESFUNC3:th 3
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
st b3 is_simple_func_in b2
holds ex b4 being disjoint_valued FinSequence of b2 st
ex b5 being FinSequence of ExtREAL st
dom b3 = union rng b4 &
dom b4 = dom b5 &
(for b6 being natural set
st b6 in dom b4
for b7 being set
st b7 in b4 . b6
holds b3 . b7 = b5 . b6) &
(for b6 being set
st b6 in dom b3
holds ex b7 being FinSequence of ExtREAL st
dom b7 = dom b5 &
(for b8 being natural set
st b8 in dom b7
holds b7 . b8 = (b5 . b8) * ((chi(b4 . b8,b1)) . b6)));
:: MESFUNC3:th 4
theorem
for b1 being set
for b2 being FinSequence of b1 holds
b2 is disjoint_valued
iff
for b3, b4 being natural set
st b3 in dom b2 & b4 in dom b2 & b3 <> b4
holds b2 . b3 misses b2 . b4;
:: MESFUNC3:th 5
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being disjoint_valued FinSequence of b3
for b5 being FinSequence of b3
st dom b5 = dom b4 &
(for b6 being natural set
st b6 in dom b5
holds b5 . b6 = b2 /\ (b4 . b6))
holds b5 is disjoint_valued FinSequence of b3;
:: MESFUNC3:th 6
theorem
for b1 being non empty set
for b2 being set
for b3, b4 being FinSequence of b1
st dom b4 = dom b3 &
(for b5 being natural set
st b5 in dom b4
holds b4 . b5 = b2 /\ (b3 . b5))
holds union rng b4 = b2 /\ union rng b3;
:: MESFUNC3:th 7
theorem
for b1 being set
for b2 being FinSequence of b1
for b3 being natural set
st b3 in dom b2
holds b2 . b3 c= union rng b2 &
(b2 . b3) /\ union rng b2 = b2 . b3;
:: MESFUNC3:th 8
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being disjoint_valued FinSequence of b2 holds
dom b4 = dom (b3 * b4);
:: MESFUNC3:th 9
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being disjoint_valued FinSequence of b2 holds
b3 . union rng b4 = Sum (b3 * b4);
:: MESFUNC3:th 10
theorem
for b1, b2 being FinSequence of ExtREAL
for b3 being Element of ExtREAL
st ((b3 = +infty or b3 = -infty) &
(ex b4 being natural set st
b4 in dom b1 & 0. <= b1 . b4) implies for b4 being natural set
st b4 in dom b1
holds 0. < b1 . b4) &
dom b1 = dom b2 &
(for b4 being natural set
st b4 in dom b2
holds b2 . b4 = b3 * (b1 . b4))
holds Sum b2 = b3 * Sum b1;
:: MESFUNC3:th 11
theorem
for b1 being FinSequence of REAL holds
b1 is FinSequence of ExtREAL;
:: MESFUNC3:prednot 1 => MESFUNC3:pred 1
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Function-like Relation of a1,ExtREAL;
let a4 be disjoint_valued FinSequence of a2;
let a5 be FinSequence of ExtREAL;
pred A4,A5 are_Re-presentation_of A3 means
dom a3 = union rng a4 &
dom a4 = dom a5 &
(for b1 being natural set
st b1 in dom a4
for b2 being set
st b2 in a4 . b1
holds a3 . b2 = a5 . b1);
end;
:: MESFUNC3:dfs 1
definiens
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Function-like Relation of a1,ExtREAL;
let a4 be disjoint_valued FinSequence of a2;
let a5 be FinSequence of ExtREAL;
To prove
a4,a5 are_Re-presentation_of a3
it is sufficient to prove
thus dom a3 = union rng a4 &
dom a4 = dom a5 &
(for b1 being natural set
st b1 in dom a4
for b2 being set
st b2 in a4 . b1
holds a3 . b2 = a5 . b1);
:: MESFUNC3:def 1
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being disjoint_valued FinSequence of b2
for b5 being FinSequence of ExtREAL holds
b4,b5 are_Re-presentation_of b3
iff
dom b3 = union rng b4 &
dom b4 = dom b5 &
(for b6 being natural set
st b6 in dom b4
for b7 being set
st b7 in b4 . b6
holds b3 . b7 = b5 . b6);
:: MESFUNC3:th 12
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
st b3 is_simple_func_in b2
holds ex b4 being disjoint_valued FinSequence of b2 st
ex b5 being FinSequence of ExtREAL st
b4,b5 are_Re-presentation_of b3;
:: MESFUNC3:th 13
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being disjoint_valued FinSequence of b2 holds
ex b4 being disjoint_valued FinSequence of b2 st
union rng b3 = union rng b4 &
(for b5 being natural set
st b5 in dom b4
holds b4 . b5 <> {} &
(ex b6 being natural set st
b6 in dom b3 & b3 . b6 = b4 . b5));
:: MESFUNC3:th 14
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
st b3 is_simple_func_in b2 &
(for b4 being set
st b4 in dom b3
holds 0. <= b3 . b4)
holds ex b4 being disjoint_valued FinSequence of b2 st
ex b5 being FinSequence of ExtREAL st
b4,b5 are_Re-presentation_of b3 &
b5 . 1 = 0. &
(for b6 being natural set
st 2 <= b6 & b6 in dom b5
holds 0. < b5 . b6 & b5 . b6 < +infty);
:: MESFUNC3:th 15
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being disjoint_valued FinSequence of b2
for b5 being FinSequence of ExtREAL
for b6 being Element of b1
st b4,b5 are_Re-presentation_of b3 & b6 in dom b3
holds ex b7 being FinSequence of ExtREAL st
dom b7 = dom b5 &
(for b8 being natural set
st b8 in dom b7
holds b7 . b8 = (b5 . b8) * ((chi(b4 . b8,b1)) . b6)) &
b3 . b6 = Sum b7;
:: MESFUNC3:th 16
theorem
for b1 being FinSequence of ExtREAL
for b2 being FinSequence of REAL
st b1 = b2
holds Sum b1 = Sum b2;
:: MESFUNC3:th 17
theorem
for b1 being FinSequence of ExtREAL
st (for b2 being natural set
st b2 in dom b1
holds 0. <= b1 . b2) &
(ex b2 being natural set st
b2 in dom b1 & b1 . b2 = +infty)
holds Sum b1 = +infty;
:: MESFUNC3:funcnot 1 => MESFUNC3:func 1
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
let a4 be Function-like Relation of a1,ExtREAL;
assume a4 is_simple_func_in a2 &
dom a4 <> {} &
(for b1 being set
st b1 in dom a4
holds 0. <= a4 . b1);
func integral(A1,A2,A3,A4) -> Element of ExtREAL means
ex b1 being disjoint_valued FinSequence of a2 st
ex b2, b3 being FinSequence of ExtREAL st
b1,b2 are_Re-presentation_of a4 &
b2 . 1 = 0. &
(for b4 being natural set
st 2 <= b4 & b4 in dom b2
holds 0. < b2 . b4 & b2 . b4 < +infty) &
dom b3 = dom b1 &
(for b4 being natural set
st b4 in dom b3
holds b3 . b4 = (b2 . b4) * ((a3 * b1) . b4)) &
it = Sum b3;
end;
:: MESFUNC3:def 2
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
st b4 is_simple_func_in b2 &
dom b4 <> {} &
(for b5 being set
st b5 in dom b4
holds 0. <= b4 . b5)
for b5 being Element of ExtREAL holds
b5 = integral(b1,b2,b3,b4)
iff
ex b6 being disjoint_valued FinSequence of b2 st
ex b7, b8 being FinSequence of ExtREAL st
b6,b7 are_Re-presentation_of b4 &
b7 . 1 = 0. &
(for b9 being natural set
st 2 <= b9 & b9 in dom b7
holds 0. < b7 . b9 & b7 . b9 < +infty) &
dom b8 = dom b6 &
(for b9 being natural set
st b9 in dom b8
holds b8 . b9 = (b7 . b9) * ((b3 * b6) . b9)) &
b5 = Sum b8;
:: MESFUNC3:th 18
theorem
for b1 being FinSequence of ExtREAL
for b2, b3 being Element of ExtREAL
st b3 = len b1 &
(for b4 being natural set
st b4 in dom b1
holds b1 . b4 = b2)
holds Sum b1 = b3 * b2;