let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,ExtREAL
for A being Element of S st f is real-valued & g is real-valued & 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,ExtREAL
for A being Element of S st f is real-valued & g is real-valued & 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,ExtREAL; :: thesis: for A being Element of S st f is real-valued & g is real-valued & 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 real-valued & g is real-valued & f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )
assume that
A1: f is real-valued and
A2: g is real-valued and
A3: f is A -measurable and
A4: ( g is A -measurable & A c= dom g ) ; :: thesis: f - g is A -measurable
A5: (- 1) (#) g is real-valued by A2, Th10;
A6: (- 1) (#) g is A -measurable by A4, MESFUNC1:37;
A7: - g is real-valued by A5, Th9;
- g is A -measurable by A6, Th9;
then f + (- g) is A -measurable by A1, A3, A7, Th7;
hence f - g is A -measurable by Th8; :: thesis: verum