let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2
for f being b₄ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2
for f being b₃ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2
for f being b₂ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable
let M1 be sigma_Measure of S1; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2
for f being b₁ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for V being Element of S2
for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable
let V be Element of S2; :: thesis: for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable
let f be E -measurable PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & E = dom f implies Integral1 (M1,|.f.|) is V -measurable )
assume that
A1: M1 is sigma_finite and
A2: E = dom f ; :: thesis: Integral1 (M1,|.f.|) is V -measurable
A3: E = dom |.f.| by A2, MESFUNC1:def 10;
|.f.| is E -measurable by A2, MESFUNC2:27;
hence Integral1 (M1,|.f.|) is V -measurable by A1, A3, MESFUN12:59; :: thesis: verum