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 E 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 ) & E = dom f & f is E -measurable holds
Integral1 (M1,f) is V -measurable
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 E 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 ) & E = dom f & f is E -measurable holds
Integral1 (M1,f) is V -measurable
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for f being PartFunc of [:X1,X2:],ExtREAL
for E 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 ) & E = dom f & f is E -measurable holds
Integral1 (M1,f) is V -measurable
let M1 be sigma_Measure of S1; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for E 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 ) & E = dom f & f is E -measurable holds
Integral1 (M1,f) is V -measurable
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for E 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 ) & E = dom f & f is E -measurable holds
Integral1 (M1,f) is V -measurable
let 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 ) & A = dom f & f is A -measurable holds
Integral1 (M1,f) is V -measurable
let V be Element of S2; :: thesis: ( M1 is sigma_finite & ( f is nonnegative or f is nonpositive ) & A = dom f & f is A -measurable implies Integral1 (M1,f) is V -measurable )
assume that
A1: M1 is sigma_finite and
A3: ( f is nonnegative or f is nonpositive ) and
A4: A = dom f and
A5: f is A -measurable ; :: thesis: Integral1 (M1,f) is V -measurable
consider I1 being Function of X2,ExtREAL such that
A6: for y being Element of X2 holds I1 . y = Integral (M1,(ProjPMap2 (f,y))) and
A7: for W being Element of S2 holds I1 is W -measurable by A1, A3, A4, A5, Lm7, Lm8;
I1 = Integral1 (M1,f) by A6, Def7;
hence Integral1 (M1,f) is V -measurable by A7; :: thesis: verum