let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,REAL
for A being Element of S st f is_measurable_on A & A c= dom f holds
abs f is_measurable_on A
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for A being Element of S st f is_measurable_on A & A c= dom f holds
abs f is_measurable_on A
let f be PartFunc of X,REAL ; :: thesis: for A being Element of S st f is_measurable_on A & A c= dom f holds
abs f is_measurable_on A
let A be Element of S; :: thesis: ( f is_measurable_on A & A c= dom f implies abs f is_measurable_on A )
assume A1: ( f is_measurable_on A & A c= dom f ) ; :: thesis: abs f is_measurable_on A
then R_EAL f is_measurable_on A by Def6;
then |.(R_EAL f).| is_measurable_on A by A1, MESFUNC2:29;
then R_EAL (abs f) is_measurable_on A by Th44;
hence abs f is_measurable_on A by Def6; :: thesis: verum