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 V82() & g is V82() & 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 V82() & g is V82() & 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 V82() & g is V82() & 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 V82() & g is V82() & f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )

assume that

A1: f is V82() and

A2: g is V82() 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 V82() by A2, Th10;

A6: (- 1) (#) g is A -measurable by A4, MESFUNC1:37;

A7: - g is V82() 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

for f, g being PartFunc of X,ExtREAL

for A being Element of S st f is V82() & g is V82() & 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 V82() & g is V82() & 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 V82() & g is V82() & 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 V82() & g is V82() & f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )

assume that

A1: f is V82() and

A2: g is V82() 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 V82() by A2, Th10;

A6: (- 1) (#) g is A -measurable by A4, MESFUNC1:37;

A7: - g is V82() 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