let X1, X2 be non empty set ; 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 M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )
let S1 be 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 M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )
let S2 be 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 M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )
let M1 be sigma_Measure of S1; for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )
let M2 be sigma_Measure of S2; for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds
( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )
let f be PartFunc of [:X1,X2:],ExtREAL; ( M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) implies ( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 ) )
assume that
A1:
M1 is sigma_finite
and
A2:
f is_integrable_on Prod_Measure (M1,M2)
; ( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )
consider E being Element of sigma (measurable_rectangles (S1,S2)) such that
A3:
( E = dom f & f is E -measurable )
by A2, MESFUNC5:def 17;
reconsider SX2 = X2 as Element of S2 by MEASURE1:7;
A4:
Integral1 (M1,|.f.|) is SX2 -measurable
by A1, A3, Th5;
( E = dom |.f.| & |.f.| is E -measurable )
by A3, MESFUNC1:def 10, MESFUNC2:27;
then A5:
Integral1 (M1,|.f.|) is V97()
by MESFUN12:66;
hence
Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|)))
by MESFUN11:31; Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0
SX2 = dom (Integral1 (M1,|.f.|))
by FUNCT_2:def 1;
hence
Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0
by A4, A5, MESFUN11:53; verum