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 A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,REAL
for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable
let f, g be PartFunc of X,REAL; :: thesis: for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable
let A be Element of S; :: thesis: ( f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )
assume that
A1: f is A -measurable and
A2: g is A -measurable and
A3: A c= dom g ; :: thesis: f - g is A -measurable
R_EAL g is A -measurable by A2;
then (- 1) (#) (R_EAL g) is A -measurable by A3, MESFUNC1:37;
then - (R_EAL g) is A -measurable by MESFUNC2:9;
then A4: R_EAL (- g) is A -measurable by Th28;
R_EAL f is A -measurable by A1;
then (R_EAL f) + (R_EAL (- g)) is A -measurable by A4, MESFUNC2:7;
then R_EAL (f - g) is A -measurable by Th23;
hence f - g is A -measurable ; :: thesis: verum