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 ) )
A1: max+ f is nonnegative by Lm1;
assume A2: 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
A3: E = dom f and
A4: f is E -measurable ;
A5: max+ f is E -measurable by A4, MESFUNC2:25;
A6: f is E /\ A -measurable by A4, MESFUNC1:30, XBOOLE_1:17;
(dom f) /\ (E /\ A) = E /\ A by A3, XBOOLE_1:17, XBOOLE_1:28;
then f | (E /\ A) is E /\ A -measurable by A6, Th42;
then (f | E) | A is E /\ A -measurable by RELAT_1:71;
then A7: f | A is E /\ A -measurable by A3, GRFUNC_1:23;
A8: integral+ (M,(max- f)) < +infty by A2;
A9: max- f is nonnegative by Lm1;
A10: integral+ (M,(max+ f)) < +infty by A2;
A11: (max+ f) | (E /\ A) = ((max+ f) | E) | A by RELAT_1:71;
A12: dom f = dom (max+ f) by MESFUNC2:def 2;
then (max+ f) | E = max+ f by A3, GRFUNC_1:23;
then A13: integral+ (M,((max+ f) | A)) <= integral+ (M,(max+ f)) by A3, A5, A12, A1, A11, Th83, XBOOLE_1:17;
then integral+ (M,(max+ (f | A))) <= integral+ (M,(max+ f)) by Th28;
then A14: integral+ (M,(max+ (f | A))) < +infty by A10, XXREAL_0:2;
A15: max- f is E -measurable by A3, A4, MESFUNC2:26;
A16: (max- f) | (E /\ A) = ((max- f) | E) | A by RELAT_1:71;
A17: dom f = dom (max- f) by MESFUNC2:def 3;
then (max- f) | E = max- f by A3, GRFUNC_1:23;
then A18: integral+ (M,((max- f) | A)) <= integral+ (M,(max- f)) by A3, A15, A17, A9, A16, Th83, XBOOLE_1:17;
then integral+ (M,(max- (f | A))) <= integral+ (M,(max- f)) by Th28;
then A19: integral+ (M,(max- (f | A))) < +infty by A8, XXREAL_0:2;
E /\ A = dom (f | A) by A3, RELAT_1:61;
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 A13, A18, A7, A14, A19, Th28; :: thesis: verum