let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1
for f being b₄ -measurable PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & E = dom f holds
Integral2 (M2,|.f.|) is U -measurable
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1
for f being b₃ -measurable PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & E = dom f holds
Integral2 (M2,|.f.|) is U -measurable
let S2 be SigmaField of X2; :: thesis: for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1
for f being b₂ -measurable PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & E = dom f holds
Integral2 (M2,|.f.|) is U -measurable
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1
for f being b₁ -measurable PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & E = dom f holds
Integral2 (M2,|.f.|) is U -measurable
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for U being Element of S1
for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & E = dom f holds
Integral2 (M2,|.f.|) is U -measurable
let U be Element of S1; :: thesis: for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & E = dom f holds
Integral2 (M2,|.f.|) is U -measurable
let f be E -measurable PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M2 is sigma_finite & E = dom f implies Integral2 (M2,|.f.|) is U -measurable )
assume that
A1: M2 is sigma_finite and
A2: E = dom f ; :: thesis: Integral2 (M2,|.f.|) is U -measurable
A3: E = dom |.f.| by A2, MESFUNC1:def 10;
|.f.| is E -measurable by A2, MESFUNC2:27;
hence Integral2 (M2,|.f.|) is U -measurable by A1, A3, MESFUN12:60; :: thesis: verum