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 f being PartFunc of [:X1,X2:],ExtREAL
for E1, E2 being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2 st M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E1 = dom f & f is_measurable_on E1 holds
Integral1 (M1,(f | E2)) is_measurable_on V
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for f being PartFunc of [:X1,X2:],ExtREAL
for E1, E2 being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2 st M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E1 = dom f & f is_measurable_on E1 holds
Integral1 (M1,(f | E2)) is_measurable_on V
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for f being PartFunc of [:X1,X2:],ExtREAL
for E1, E2 being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2 st M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E1 = dom f & f is_measurable_on E1 holds
Integral1 (M1,(f | E2)) is_measurable_on V
let M1 be sigma_Measure of S1; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for E1, E2 being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2 st M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E1 = dom f & f is_measurable_on E1 holds
Integral1 (M1,(f | E2)) is_measurable_on V
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for E1, E2 being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2 st M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E1 = dom f & f is_measurable_on E1 holds
Integral1 (M1,(f | E2)) is_measurable_on V
let E, A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for V being Element of S2 st M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is_measurable_on E holds
Integral1 (M1,(f | A)) is_measurable_on V
let V be Element of S2; :: thesis: ( M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is_measurable_on E implies Integral1 (M1,(f | A)) is_measurable_on V )
assume that
A1: M1 is sigma_finite and
A2: ( f is nonnegative or f is nonpositive ) and
A3: E = dom f and
A4: f is_measurable_on E ; :: thesis: Integral1 (M1,(f | A)) is_measurable_on V
A5: dom (f | A) = E /\ A by A3, RELAT_1:61;
A6: (dom f) /\ (E /\ A) = E /\ A by A3, XBOOLE_1:17, XBOOLE_1:28;
f is_measurable_on E /\ A by A4, XBOOLE_1:17, MESFUNC1:30;
then f | (E /\ A) is_measurable_on E /\ A by A6, MESFUNC5:42;
then (f | E) | A is_measurable_on E /\ A by RELAT_1:71;
hence Integral1 (M1,(f | A)) is_measurable_on V by A1, A2, A3, A5, MESFUNC5:15, MESFUN11:1, Th59; :: thesis: verum