let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for B, A being Element of S st B c= A & f is_measurable_on A holds
f is_measurable_on B
let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX
for B, A being Element of S st B c= A & f is_measurable_on A holds
f is_measurable_on B
let f be PartFunc of X,COMPLEX ; :: thesis: for B, A being Element of S st B c= A & f is_measurable_on A holds
f is_measurable_on B
let B, A be Element of S; :: thesis: ( B c= A & f is_measurable_on A implies f is_measurable_on B )
assume A1: B c= A ; :: thesis: ( not f is_measurable_on A or f is_measurable_on B )
assume f is_measurable_on A ; :: thesis: f is_measurable_on B
then ( Re f is_measurable_on A & Im f is_measurable_on A ) by Def3;
then ( Re f is_measurable_on B & Im f is_measurable_on B ) by A1, MESFUNC6:16;
hence f is_measurable_on B by Def3; :: thesis: verum