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 x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
(Y-vol (E,M2)) . x = Integral (M2,(chi ((Measurable-X-section (E,x)),X2)))
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
(Y-vol (E,M2)) . x = Integral (M2,(chi ((Measurable-X-section (E,x)),X2)))
let S2 be SigmaField of X2; :: thesis: for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
(Y-vol (E,M2)) . x = Integral (M2,(chi ((Measurable-X-section (E,x)),X2)))
let M2 be sigma_Measure of S2; :: thesis: for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
(Y-vol (E,M2)) . x = Integral (M2,(chi ((Measurable-X-section (E,x)),X2)))
let x be Element of X1; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st M2 is sigma_finite holds
(Y-vol (E,M2)) . x = Integral (M2,(chi ((Measurable-X-section (E,x)),X2)))
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( M2 is sigma_finite implies (Y-vol (A,M2)) . x = Integral (M2,(chi ((Measurable-X-section (A,x)),X2))) )
assume M2 is sigma_finite ; :: thesis: (Y-vol (A,M2)) . x = Integral (M2,(chi ((Measurable-X-section (A,x)),X2)))
then (Y-vol (A,M2)) . x = M2 . (Measurable-X-section (A,x)) by MEASUR11:def 13;
hence (Y-vol (A,M2)) . x = Integral (M2,(chi ((Measurable-X-section (A,x)),X2))) by MESFUNC9:14; :: thesis: verum