let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A, B 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,ExtREAL
for A, B 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,ExtREAL ; :: thesis: for A, B being Element of S st B c= A & f is_measurable_on A holds
f is_measurable_on B
let A, B be Element of S; :: thesis: ( B c= A & f is_measurable_on A implies f is_measurable_on B )
assume that
A1: B c= A and
A2: f is_measurable_on A ; :: thesis: f is_measurable_on B
for r being real number holds B /\ (less_dom f,(R_EAL r)) in S

proof

let r be real number ; :: thesis: B /\ (less_dom f,(R_EAL r)) in S
A3: A /\ (less_dom f,(R_EAL r)) in S by A2, Def17;
B /\ (A /\ (less_dom f,(R_EAL r))) = (B /\ A) /\ (less_dom f,(R_EAL r)) by XBOOLE_1:16
.= B /\ (less_dom f,(R_EAL r)) by A1, XBOOLE_1:28 ;
hence B /\ (less_dom f,(R_EAL r)) in S by A3, MEASURE1:19; :: thesis: verum

end;

hence f is_measurable_on B by Def17; :: thesis: verum