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 and
A2: g is_measurable_on A and
A3: A c= dom g ; :: thesis: f - g is_measurable_on A
R_EAL g is_measurable_on A by A2, Def6;
then (- 1) (#) (R_EAL g) is_measurable_on A by A3, MESFUNC1:41;
then - (R_EAL g) is_measurable_on A by MESFUNC2:11;
then A4: R_EAL (- g) is_measurable_on A by Th28;
R_EAL f is_measurable_on A by A1, Def6;
then (R_EAL f) + (R_EAL (- g)) is_measurable_on A by A4, MESFUNC2:7;
then R_EAL (f - g) is_measurable_on A by Th23;
hence f - g is_measurable_on A by Def6; :: thesis: verum