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 M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b₅ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & dom f = E holds
( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b₄ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & dom f = E holds
( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b₃ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & dom f = E holds
( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b₂ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & dom f = E holds
( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b₁ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & dom f = E holds
( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & dom f = E holds
( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
let f be E -measurable PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & M2 is sigma_finite & dom f = E implies ( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) ) )
assume that
A1: M1 is sigma_finite and
A2: M2 is sigma_finite and
A3: dom f = E ; :: thesis: ( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) )
( E = dom |.f.| & |.f.| is E -measurable ) by A3, MESFUNC1:def 10, MESFUNC2:27;
hence ( Integral (M1,(Integral2 (M2,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) & Integral (M2,(Integral1 (M1,|.f.|))) = Integral ((Prod_Measure (M1,M2)),|.f.|) ) by A1, A2, MESFUN12:84; :: thesis: verum