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 st f is_integrable_on M holds
( 0 <= integral+ M,(max+ f) & 0 <= integral+ M,(max- f) & -infty < Integral M,f & Integral M,f < +infty )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL st f is_integrable_on M holds
( 0 <= integral+ M,(max+ f) & 0 <= integral+ M,(max- f) & -infty < Integral M,f & Integral M,f < +infty )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL st f is_integrable_on M holds
( 0 <= integral+ M,(max+ f) & 0 <= integral+ M,(max- f) & -infty < Integral M,f & Integral M,f < +infty )
let f be PartFunc of X,ExtREAL ; :: thesis: ( f is_integrable_on M implies ( 0 <= integral+ M,(max+ f) & 0 <= integral+ M,(max- f) & -infty < Integral M,f & Integral M,f < +infty ) )
assume A1: f is_integrable_on M ; :: thesis: ( 0 <= integral+ M,(max+ f) & 0 <= integral+ M,(max- f) & -infty < Integral M,f & Integral M,f < +infty )
consider A being Element of S such that
A2: ( A = dom f & f is_measurable_on A ) by A1, Def17;
A3: ( dom f = dom (max+ f) & dom f = dom (max- f) ) by MESFUNC2:def 2, MESFUNC2:def 3;
A4: ( max+ f is nonnegative & max- f is nonnegative ) by Lm1;
A5: ( max+ f is_measurable_on A & max- f is_measurable_on A ) by A2, MESFUNC2:27, MESFUNC2:28;
then A6: ( -infty <> integral+ M,(max+ f) & -infty <> integral+ M,(max- f) ) by A2, A3, A4, Th85;
A7: ( integral+ M,(max+ f) <> +infty & integral+ M,(max- f) <> +infty ) by A1, Def17;
then reconsider maxf1 = integral+ M,(max+ f) as Real by A6, XXREAL_0:14;
reconsider maxf2 = integral+ M,(max- f) as Real by A6, A7, XXREAL_0:14;
(integral+ M,(max+ f)) - (integral+ M,(max- f)) = maxf1 - maxf2 by SUPINF_2:5;
hence ( 0 <= integral+ M,(max+ f) & 0 <= integral+ M,(max- f) & -infty < Integral M,f & Integral M,f < +infty ) by A2, A3, A4, A5, Th85, XXREAL_0:9, XXREAL_0:12; :: thesis: verum