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 E being Element of sigma (measurable_rectangles (S1,S2))
for f being b5 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) holds
f is_integrable_on Prod_Measure (M1,M2)
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 E being Element of sigma (measurable_rectangles (S1,S2))
for f being b4 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) holds
f is_integrable_on Prod_Measure (M1,M2)
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b3 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) holds
f is_integrable_on Prod_Measure (M1,M2)
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b2 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) holds
f is_integrable_on Prod_Measure (M1,M2)
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for f being b1 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) holds
f is_integrable_on Prod_Measure (M1,M2)
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) holds
f is_integrable_on Prod_Measure (M1,M2)
let f be E -measurable PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & M2 is sigma_finite & E = dom f & ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) implies f is_integrable_on Prod_Measure (M1,M2) )
assume that
A1: M1 is sigma_finite and
A2: M2 is sigma_finite and
A3: E = dom f ; :: thesis: ( ( not Integral (M2,(Integral1 (M1,|.f.|))) < +infty & not Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) or f is_integrable_on Prod_Measure (M1,M2) )
set M = Prod_Measure (M1,M2);
A5: E = dom |.f.| by A3, MESFUNC1:def 10;
then A6: E = dom (max- |.f.|) by MESFUNC2:def 3;
A7: |.f.| is E -measurable by A3, MESFUNC2:27;
assume ( Integral (M2,(Integral1 (M1,|.f.|))) < +infty or Integral (M1,(Integral2 (M2,|.f.|))) < +infty ) ; :: thesis: f is_integrable_on Prod_Measure (M1,M2)
then Integral ((Prod_Measure (M1,M2)),|.f.|) < +infty by A1, A2, A5, A7, MESFUN12:84;
then integral+ ((Prod_Measure (M1,M2)),|.f.|) < +infty by A3, A5, MESFUNC2:27, MESFUNC5:88;
then A8: integral+ ((Prod_Measure (M1,M2)),(max+ |.f.|)) < +infty by MESFUN11:31;
now :: thesis: for z being Element of [:X1,X2:] st z in dom (max- |.f.|) holds
(max- |.f.|) . z = 0
let z be Element of [:X1,X2:]; :: thesis: ( z in dom (max- |.f.|) implies (max- |.f.|) . z = 0 )
assume z in dom (max- |.f.|) ; :: thesis: (max- |.f.|) . z = 0
(max+ |.f.|) . z = |.f.| . z by MESFUN11:31;
hence (max- |.f.|) . z = 0 by MESFUNC2:19; :: thesis: verum
end;
then integral+ ((Prod_Measure (M1,M2)),(max- |.f.|)) = 0 by A5, A6, A7, MESFUNC2:26, MESFUNC5:87;
then |.f.| is_integrable_on Prod_Measure (M1,M2) by A3, A5, A8, MESFUNC2:27, MESFUNC5:def 17;
hence f is_integrable_on Prod_Measure (M1,M2) by A3, MESFUNC5:100; :: thesis: verum