let X be non empty set ; :: thesis: for S being SigmaField of X

for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f is A -measurable

let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f is A -measurable

let f be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is_simple_func_in S holds

f is A -measurable

let A be Element of S; :: thesis: ( f is_simple_func_in S implies f is A -measurable )

assume A1: f is_simple_func_in S ; :: thesis: f is A -measurable

then Im f is_simple_func_in S by MESFUN7C:43;

then A2: Im f is A -measurable by MESFUNC6:50;

Re f is_simple_func_in S by A1, MESFUN7C:43;

then Re f is A -measurable by MESFUNC6:50;

hence f is A -measurable by A2, MESFUN6C:def 1; :: thesis: verum

for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f is A -measurable

let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX

for A being Element of S st f is_simple_func_in S holds

f is A -measurable

let f be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is_simple_func_in S holds

f is A -measurable

let A be Element of S; :: thesis: ( f is_simple_func_in S implies f is A -measurable )

assume A1: f is_simple_func_in S ; :: thesis: f is A -measurable

then Im f is_simple_func_in S by MESFUN7C:43;

then A2: Im f is A -measurable by MESFUNC6:50;

Re f is_simple_func_in S by A1, MESFUN7C:43;

then Re f is A -measurable by MESFUNC6:50;

hence f is A -measurable by A2, MESFUN6C:def 1; :: thesis: verum