let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,REAL
for A being Element of S st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
f - g is_measurable_on A
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,REAL
for A being Element of S st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
f - g is_measurable_on A
let f, g be PartFunc of X,REAL ; :: thesis: for A being Element of S st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
f - g is_measurable_on A
let A be Element of S; :: thesis: ( f is_measurable_on A & g is_measurable_on A & A c= dom g implies f - g is_measurable_on A )
assume that
A1: ( f is_measurable_on A & g is_measurable_on A ) and
A2: A c= dom g ; :: thesis: f - g is_measurable_on A
A3: ( R_EAL f is_measurable_on A & R_EAL g is_measurable_on A ) by A1, Def6;
then (- 1) (#) (R_EAL g) is_measurable_on A by A2, MESFUNC1:41;
then - (R_EAL g) is_measurable_on A by MESFUNC2:11;
then R_EAL (- g) is_measurable_on A by Th28;
then (R_EAL f) + (R_EAL (- g)) is_measurable_on A by A3, MESFUNC2:7;
then R_EAL (f - g) is_measurable_on A by Th23;
hence f - g is_measurable_on A by Def6; :: thesis: verum