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
for r being Real st f is_measurable_on A & A c= dom f holds
r (#) 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
for r being Real st f is_measurable_on A & A c= dom f holds
r (#) f is_measurable_on A
let f be PartFunc of X,REAL ; :: thesis: for A being Element of S
for r being Real st f is_measurable_on A & A c= dom f holds
r (#) f is_measurable_on A
let A be Element of S; :: thesis: for r being Real st f is_measurable_on A & A c= dom f holds
r (#) f is_measurable_on A
let r be Real; :: thesis: ( f is_measurable_on A & A c= dom f implies r (#) f is_measurable_on A )
assume A1: ( f is_measurable_on A & A c= dom f ) ; :: thesis: r (#) f is_measurable_on A
then R_EAL f is_measurable_on A by Def6;
then r (#) (R_EAL f) is_measurable_on A by A1, MESFUNC1:41;
then R_EAL (r (#) f) is_measurable_on A by Th20;
hence r (#) f is_measurable_on A by Def6; :: thesis: verum