let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A being Element of S st f is_integrable_on M holds
( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A being Element of S st f is_integrable_on M holds
( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for A being Element of S st f is_integrable_on M holds
( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M )
let f be PartFunc of X,ExtREAL ; :: thesis: for A being Element of S st f is_integrable_on M holds
( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M )
let A be Element of S; :: thesis: ( f is_integrable_on M implies ( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M ) )
assume A1: f is_integrable_on M ; :: thesis: ( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M )
then consider E being Element of S such that
A2: ( E = dom f & f is_measurable_on E ) by Def17;
A3: ( max+ f is_measurable_on E & max- f is_measurable_on E ) by A2, MESFUNC2:27, MESFUNC2:28;
A4: ( dom f = dom (max+ f) & dom f = dom (max- f) ) by MESFUNC2:def 2, MESFUNC2:def 3;
A5: ( max+ f is nonnegative & max- f is nonnegative ) by Lm1;
A6: E /\ A c= E by XBOOLE_1:17;
( (max+ f) | (E /\ A) = ((max+ f) | E) | A & (max- f) | (E /\ A) = ((max- f) | E) | A ) by RELAT_1:100;
then ( (max+ f) | (E /\ A) = (max+ f) | A & (max+ f) | E = max+ f & (max- f) | (E /\ A) = (max- f) | A & (max- f) | E = max- f ) by A2, A4, GRFUNC_1:64;
then A7: ( integral+ M,((max+ f) | A) <= integral+ M,(max+ f) & integral+ M,((max- f) | A) <= integral+ M,(max- f) ) by A2, A3, A4, A5, Th89, XBOOLE_1:17;
A8: E /\ A = dom (f | A) by A2, RELAT_1:90;
( (dom f) /\ (E /\ A) = E /\ A & f is_measurable_on E /\ A ) by A2, A6, MESFUNC1:34, XBOOLE_1:28;
then f | (E /\ A) is_measurable_on E /\ A by Th48;
then (f | E) | A is_measurable_on E /\ A by RELAT_1:100;
then A9: f | A is_measurable_on E /\ A by A2, GRFUNC_1:64;
A10: ( integral+ M,(max+ f) < +infty & integral+ M,(max- f) < +infty ) by A1, Def17;
( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) ) by A7, Th34;
then ( integral+ M,(max+ (f | A)) < +infty & integral+ M,(max- (f | A)) < +infty ) by A10, XXREAL_0:2;
hence ( integral+ M,(max+ (f | A)) <= integral+ M,(max+ f) & integral+ M,(max- (f | A)) <= integral+ M,(max- f) & f | A is_integrable_on M ) by A7, A8, A9, Def17, Th34; :: thesis: verum