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 M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

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 M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

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 M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) implies ( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty ) )

assume that

A1: M1 is sigma_finite and

A2: M2 is sigma_finite and

A3: f is_integrable_on Prod_Measure (M1,M2) ; :: thesis: ( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

consider E being Element of sigma (measurable_rectangles (S1,S2)) such that

A4: ( E = dom f & f is E -measurable ) by A3, MESFUNC5:def 17;

A5: |.f.| is_integrable_on Prod_Measure (M1,M2) by A3, A4, MESFUNC5:100;

E = dom |.f.| by A4, MESFUNC1:def 10;

then Integral ((Prod_Measure (M1,M2)),|.f.|) = integral+ ((Prod_Measure (M1,M2)),|.f.|) by A4, MESFUNC2:27, MESFUNC5:88

.= integral+ ((Prod_Measure (M1,M2)),(max+ |.f.|)) by MESFUN11:31 ;

then Integral ((Prod_Measure (M1,M2)),|.f.|) < +infty by A5, MESFUNC5:def 17;

hence ( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty ) by A1, A2, A4, Th1; :: thesis: verum

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 & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

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 M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

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 M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & M2 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) implies ( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty ) )

assume that

A1: M1 is sigma_finite and

A2: M2 is sigma_finite and

A3: f is_integrable_on Prod_Measure (M1,M2) ; :: thesis: ( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty )

consider E being Element of sigma (measurable_rectangles (S1,S2)) such that

A4: ( E = dom f & f is E -measurable ) by A3, MESFUNC5:def 17;

A5: |.f.| is_integrable_on Prod_Measure (M1,M2) by A3, A4, MESFUNC5:100;

E = dom |.f.| by A4, MESFUNC1:def 10;

then Integral ((Prod_Measure (M1,M2)),|.f.|) = integral+ ((Prod_Measure (M1,M2)),|.f.|) by A4, MESFUNC2:27, MESFUNC5:88

.= integral+ ((Prod_Measure (M1,M2)),(max+ |.f.|)) by MESFUN11:31 ;

then Integral ((Prod_Measure (M1,M2)),|.f.|) < +infty by A5, MESFUNC5:def 17;

hence ( Integral (M1,(Integral2 (M2,|.f.|))) < +infty & Integral (M2,(Integral1 (M1,|.f.|))) < +infty ) by A1, A2, A4, Th1; :: thesis: verum