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)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
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)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
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)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
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)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( M1 is sigma_finite & M2 is sigma_finite implies ( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) ) )
assume that
A1: M1 is sigma_finite and
A2: M2 is sigma_finite ; :: thesis: ( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
X-vol (E,M1) = Integral1 (M1,(chi (E,[:X1,X2:]))) by A1, Th64;
hence A4: Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) by A1, A2, Th77; :: thesis: ( Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
A5: Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = (Prod_Measure (M1,M2)) . E by MESFUNC9:14
.= Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) by MESFUNC9:14 ;
hence Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) by A4, Th79; :: thesis: ( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )
Y-vol (E,M2) = Integral2 (M2,(chi (E,[:X1,X2:]))) by A2, Th65;
hence Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) by A1, A2, Th77; :: thesis: Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E))))
hence Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) by A5, Th79; :: thesis: verum