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 b_{4} -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 b_{3} -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 b_{2} -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 b_{1} -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

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 b

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 b

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 b

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 b

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