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 f being PartFunc of [:X1,X2:],ExtREAL
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1 st M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is E -measurable holds
Integral2 (M2,f) is U -measurable
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1 st M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is E -measurable holds
Integral2 (M2,f) is U -measurable
let S2 be SigmaField of X2; :: thesis: for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1 st M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is E -measurable holds
Integral2 (M2,f) is U -measurable
let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1 st M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is E -measurable holds
Integral2 (M2,f) is U -measurable
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for U being Element of S1 st M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & E = dom f & f is E -measurable holds
Integral2 (M2,f) is U -measurable
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for U being Element of S1 st M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & A = dom f & f is A -measurable holds
Integral2 (M2,f) is U -measurable
let U be Element of S1; :: thesis: ( M2 is sigma_finite & ( f is nonnegative or f is nonpositive ) & A = dom f & f is A -measurable implies Integral2 (M2,f) is U -measurable )
assume that
A1: M2 is sigma_finite and
A3: ( f is nonnegative or f is nonpositive ) and
A4: A = dom f and
A5: f is A -measurable ; :: thesis: Integral2 (M2,f) is U -measurable
consider I2 being Function of X1,ExtREAL such that
A6: for x being Element of X1 holds I2 . x = Integral (M2,(ProjPMap1 (f,x))) and
A7: for W being Element of S1 holds I2 is W -measurable by A1, A3, A4, A5, Lm9, Lm10;
I2 = Integral2 (M2,f) by A6, Def8;
hence Integral2 (M2,f) is U -measurable by A7; :: thesis: verum