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 f being PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 )
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 f being PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 )
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 )
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 )
let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL st M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 )
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) implies ( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 ) )
assume that
A1: M2 is sigma_finite and
A2: f is_integrable_on Prod_Measure (M1,M2) ; :: thesis: ( Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) & Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 )
consider A being Element of sigma (measurable_rectangles (S1,S2)) such that
A3: ( A = dom f & f is A -measurable ) by A2, MESFUNC5:def 17;
reconsider SX1 = X1 as Element of S1 by MEASURE1:7;
A4: Integral2 (M2,|.f.|) is SX1 -measurable by A1, A3, Th4;
( A = dom |.f.| & |.f.| is A -measurable ) by A3, MESFUNC1:def 10, MESFUNC2:27;
then A5: Integral2 (M2,|.f.|) is nonnegative by MESFUN12:66;
hence Integral (M1,(max+ (Integral2 (M2,|.f.|)))) = Integral (M1,(Integral2 (M2,|.f.|))) by MESFUN11:31; :: thesis: Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0
SX1 = dom (Integral2 (M2,|.f.|)) by FUNCT_2:def 1;
hence Integral (M1,(max- (Integral2 (M2,|.f.|)))) = 0 by A4, A5, MESFUN11:53; :: thesis: verum