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 ex A being Element of S st
( A = dom f & f is_measurable_on A ) holds
( f is_integrable_on M iff 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 ex A being Element of S st
( A = dom f & f is_measurable_on A ) holds
( f is_integrable_on M iff abs f is_integrable_on M )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st ex A being Element of S st
( A = dom f & f is_measurable_on A ) holds
( f is_integrable_on M iff abs f is_integrable_on M )
let f be PartFunc of X,REAL ; :: thesis: ( ex A being Element of S st
( A = dom f & f is_measurable_on A ) implies ( f is_integrable_on M iff abs f is_integrable_on M ) )
assume ex A being Element of S st
( A = dom f & f is_measurable_on A ) ; :: thesis: ( f is_integrable_on M iff abs f is_integrable_on M )
then consider A being Element of S such that
A1: ( A = dom f & f is_measurable_on A ) ;
( A = dom (R_EAL f) & R_EAL f is_measurable_on A ) by A1, Def6;
then ( R_EAL f is_integrable_on M iff |.(R_EAL f).| is_integrable_on M ) by MESFUNC5:106;
then ( f is_integrable_on M iff R_EAL (abs f) is_integrable_on M ) by Def9, Th1;
hence ( f is_integrable_on M iff abs f is_integrable_on M ) by Def9; :: thesis: verum