
theorem Th31:
  for X be non empty set, S be SigmaField of X, f,g be PartFunc of
  X,ExtREAL, A be Element of S st f is without-infty & g is without-infty & f
  is A-measurable & g is A-measurable holds f+g is A-measurable
proof
  let X be non empty set, S be SigmaField of X, f,g be PartFunc of X,ExtREAL,
  A be Element of S;
  assume that
A1: f is without-infty and
A2: g is without-infty and
A3: f is A-measurable and
A4: g is A-measurable;
  for r be Real holds A /\ less_dom(f+g, r) in S
  proof
    let r be Real;
    consider F being Function of RAT,S such that
A5: for p being Rational holds F.p = A /\ less_dom(f, p) /\ (A /\
    less_dom(g,(r-(p qua Complex)))) by A3,A4,MESFUNC2:6;
    ex G be sequence of S st rng F = rng G by MESFUNC1:5,MESFUNC2:5;
    then
A6: rng F is N_Sub_set_fam of X by MEASURE1:23;
    A /\ less_dom(f+g, r) = union rng F by A1,A2,A5,Th18;
    hence thesis by A6,MEASURE1:def 5;
  end;
  hence thesis by MESFUNC1:def 16;
end;
