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_measurable_on A
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_measurable_on A
let f be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is_simple_func_in S holds
f is_measurable_on A
let A be Element of S; :: thesis: ( f is_simple_func_in S implies f is_measurable_on A )
assume A1: f is_simple_func_in S ; :: thesis: f is_measurable_on A
then Im f is_simple_func_in S by MESFUN7C:43;
then A2: Im f is_measurable_on A by MESFUNC6:50;
Re f is_simple_func_in S by A1, MESFUN7C:43;
then Re f is_measurable_on A by MESFUNC6:50;
hence f is_measurable_on A by A2, MESFUN6C:def 1; :: thesis: verum