let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,ExtREAL
for A being Element of S st f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A holds
f + g is_measurable_on A
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,ExtREAL
for A being Element of S st f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A holds
f + g is_measurable_on A
let f, g be PartFunc of X,ExtREAL ; :: thesis: for A being Element of S st f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A holds
f + g is_measurable_on A
let A be Element of S; :: thesis: ( f is real-valued & g is real-valued & f is_measurable_on A & g is_measurable_on A implies f + g is_measurable_on A )
assume that
A1: ( f is real-valued & g is real-valued ) and
A2: ( f is_measurable_on A & g is_measurable_on A ) ; :: thesis: f + g is_measurable_on A
A3: for r being real number holds A /\ (less_dom (f + g),(R_EAL r)) in S

proof

let r be real number ; :: thesis: A /\ (less_dom (f + g),(R_EAL r)) in S
reconsider r = r as Real by XREAL_0:def 1;
consider F being Function of RAT ,S such that
A4: for p being Rational holds F . p = (A /\ (less_dom f,(R_EAL p))) /\ (A /\ (less_dom g,(R_EAL (r - p)))) by A2, Th6;
consider G being Function of NAT ,S such that
A5: rng F = rng G by Th5, MESFUNC1:5;
A6: A /\ (less_dom (f + g),(R_EAL r)) = union (rng G) by A1, A4, A5, Th3;
thus A /\ (less_dom (f + g),(R_EAL r)) in S by A6; :: thesis: verum

end;

thus f + g is_measurable_on A by A3, MESFUNC1:def 17; :: thesis: verum