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,REAL st f is_integrable_on M holds
( Integral M,f in REAL & Integral M,(abs f) in REAL & abs f is_integrable_on M )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_integrable_on M holds
( Integral M,f in REAL & Integral M,(abs f) in REAL & abs f is_integrable_on M )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st f is_integrable_on M holds
( Integral M,f in REAL & Integral M,(abs f) in REAL & abs f is_integrable_on M )
let f be PartFunc of X,REAL ; :: thesis: ( f is_integrable_on M implies ( Integral M,f in REAL & Integral M,(abs f) in REAL & abs f is_integrable_on M ) )
assume A1: f is_integrable_on M ; :: thesis: ( Integral M,f in REAL & Integral M,(abs f) in REAL & abs f is_integrable_on M )
then A2: ( -infty < Integral M,f & Integral M,f < +infty ) by MESFUNC6:90;
R_EAL f is_integrable_on M by A1, MESFUNC6:def 9;
then consider A being Element of S such that
A3: A = dom (R_EAL f) and
A4: R_EAL f is_measurable_on A by MESFUNC5:def 17;
A5: f is_measurable_on A by A4, MESFUNC6:def 6;
then abs f is_integrable_on M by A1, A3, MESFUNC6:94;
then ( -infty < Integral M,(abs f) & Integral M,(abs f) < +infty ) by MESFUNC6:90;
hence ( Integral M,f in REAL & Integral M,(abs f) in REAL & abs f is_integrable_on M ) by A1, A2, A3, A5, MESFUNC6:94, XXREAL_0:14; :: thesis: verum