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 that
A1: f is_measurable_on A and
A2: A c= dom f ; :: thesis: abs f is_measurable_on A
R_EAL f is_measurable_on A by A1, Def6;
then |.(R_EAL f).| is_measurable_on A by A2, MESFUNC2:27;
then R_EAL (abs f) is_measurable_on A by Th44;
hence abs f is_measurable_on A by Def6; :: thesis: verum