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

proof

let r be real number ; :: thesis: (A \/ B) /\ (less_dom f,(R_EAL r)) in S
A3: A /\ (less_dom f,(R_EAL r)) in S by A1, Def17;
B /\ (less_dom f,(R_EAL r)) in S by A2, Def17;
then (A /\ (less_dom f,(R_EAL r))) \/ (B /\ (less_dom f,(R_EAL r))) in S by A3, FINSUB_1:def 1;
hence (A \/ B) /\ (less_dom f,(R_EAL r)) in S by XBOOLE_1:23; :: thesis: verum

end;

hence f is_measurable_on A \/ B by Def17; :: thesis: verum