Article MESFUNC4, MML version 4.99.1005
:: MESFUNC4:th 1
theorem
for b1, b2, b3 being FinSequence of ExtREAL
st (for b4 being natural set
st b4 in dom b1
holds 0. <= b1 . b4) &
(for b4 being natural set
st b4 in dom b2
holds 0. <= b2 . b4) &
dom b1 = dom b2 &
b3 = b1 + b2
holds Sum b3 = (Sum b1) + Sum b2;
:: MESFUNC4:th 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 natural set
for b5 being Function-like Relation of b1,ExtREAL
for b6 being disjoint_valued FinSequence of b2
for b7, b8 being FinSequence of ExtREAL
st b5 is_simple_func_in b2 &
proj1 b5 <> {} &
(for b9 being set
st b9 in proj1 b5
holds 0. <= b5 . b9) &
b6,b7 are_Re-presentation_of b5 &
dom b8 = dom b6 &
(for b9 being natural set
st b9 in dom b8
holds b8 . b9 = (b7 . b9) * ((b3 * b6) . b9)) &
len b6 = b4
holds integral(b1,b2,b3,b5) = Sum b8;
:: MESFUNC4: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
for b4 being sigma_Measure of b2
for b5 being disjoint_valued FinSequence of b2
for b6, b7 being FinSequence of ExtREAL
st b3 is_simple_func_in b2 &
proj1 b3 <> {} &
(for b8 being set
st b8 in proj1 b3
holds 0. <= b3 . b8) &
b5,b6 are_Re-presentation_of b3 &
dom b7 = dom b5 &
(for b8 being natural set
st b8 in dom b7
holds b7 . b8 = (b6 . b8) * ((b4 * b5) . b8))
holds integral(b1,b2,b4,b3) = Sum b7;
:: MESFUNC4:th 4
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 sigma_Measure of b2
st b3 is_simple_func_in b2 &
proj1 b3 <> {} &
(for b5 being set
st b5 in proj1 b3
holds 0. <= b3 . b5)
holds ex b5 being disjoint_valued FinSequence of b2 st
ex b6, b7 being FinSequence of ExtREAL st
b5,b6 are_Re-presentation_of b3 &
dom b7 = dom b5 &
(for b8 being natural set
st b8 in dom b7
holds b7 . b8 = (b6 . b8) * ((b4 * b5) . b8)) &
integral(b1,b2,b4,b3) = Sum b7;
:: MESFUNC4:th 5
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, b5 being Function-like Relation of b1,ExtREAL
st b4 is_simple_func_in b2 &
proj1 b4 <> {} &
(for b6 being set
st b6 in proj1 b4
holds 0. <= b4 . b6) &
b5 is_simple_func_in b2 &
proj1 b5 = proj1 b4 &
(for b6 being set
st b6 in proj1 b5
holds 0. <= b5 . b6)
holds b4 + b5 is_simple_func_in b2 &
proj1 (b4 + b5) <> {} &
(for b6 being set
st b6 in proj1 (b4 + b5)
holds 0. <= (b4 + b5) . b6) &
integral(b1,b2,b3,b4 + b5) = (integral(b1,b2,b3,b4)) + integral(b1,b2,b3,b5);
:: MESFUNC4:th 6
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, b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of ExtREAL
st b4 is_simple_func_in b2 &
proj1 b4 <> {} &
(for b7 being set
st b7 in proj1 b4
holds 0. <= b4 . b7) &
0. <= b6 &
b6 < +infty &
proj1 b5 = proj1 b4 &
(for b7 being set
st b7 in proj1 b5
holds b5 . b7 = b6 * (b4 . b7))
holds integral(b1,b2,b3,b5) = b6 * integral(b1,b2,b3,b4);