let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st ex A being Element of S st
( A = (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st ex A being Element of S st
( A = (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) )
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st ex A being Element of S st
( A = (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) )
let f, g be PartFunc of X,REAL ; :: thesis: ( ex A being Element of S st
( A = (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative implies ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) ) )
assume that
A1: ex A being Element of S st
( A = (dom f) /\ (dom g) & f is_measurable_on A & g is_measurable_on A ) and
A2: f is_integrable_on M and
A3: g is_integrable_on M and
A4: g - f is nonnegative ; :: thesis: ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) )
set h = (- 1) (#) f;
(- 1) (#) f is_integrable_on M by A2, MESFUNC6:102;
then consider E being Element of S such that
A5: E = (dom ((- 1) (#) f)) /\ (dom g) and
A6: Integral M,(((- 1) (#) f) + g) = (Integral M,(((- 1) (#) f) | E)) + (Integral M,(g | E)) by A3, MESFUNC6:101;
A7: f | E is_integrable_on M by A2, MESFUNC6:91;
then A8: Integral M,(f | E) < +infty by MESFUNC6:90;
-infty < Integral M,(f | E) by A7, MESFUNC6:90;
then reconsider c1 = Integral M,(f | E) as Real by A8, XXREAL_0:14;
A9: (R_EAL (- 1)) * (Integral M,(f | E)) = (- 1) * c1 by EXTREAL1:13;
A10: g | E is_integrable_on M by A3, MESFUNC6:91;
then A11: Integral M,(g | E) < +infty by MESFUNC6:90;
-infty < Integral M,(g | E) by A10, MESFUNC6:90;
then reconsider c2 = Integral M,(g | E) as Real by A11, XXREAL_0:14;
take E ; :: thesis: ( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) )
A12: ((- 1) (#) f) | E = (- 1) (#) (f | E) by Th41;
consider A being Element of S such that
A13: A = (dom f) /\ (dom g) and
A14: f is_measurable_on A and
A15: g is_measurable_on A by A1;
dom ((- 1) (#) f) = dom f by VALUED_1:def 5;
then A16: dom (((- 1) (#) f) + g) = A by A13, VALUED_1:def 1;
(- 1) (#) f is_measurable_on A by A13, A14, MESFUNC6:21, XBOOLE_1:17;
then 0 <= (Integral M,(((- 1) (#) f) | E)) + (Integral M,(g | E)) by A4, A6, A15, A16, MESFUNC6:26, MESFUNC6:84;
then 0 <= ((R_EAL (- 1)) * (Integral M,(f | E))) + (Integral M,(g | E)) by A7, A12, MESFUNC6:102;
then 0 <= (- c1) + c2 by A9, SUPINF_2:1;
then 0 + c1 <= ((- c1) + c2) + c1 by XREAL_1:8;
hence ( E = (dom f) /\ (dom g) & Integral M,(f | E) <= Integral M,(g | E) ) by A5, VALUED_1:def 5; :: thesis: verum