
theorem Th44:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, A be Element of S, f,g being PartFunc of X,ExtREAL st A c= dom f /\ dom g
  & f is A-measurable & g is A-measurable & f is without-infty & g is
  without-infty holds max-(f+g) + max+f is A-measurable
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be
  Element of S, f,g be PartFunc of X,ExtREAL;
  assume that
A1: A c= dom f /\ dom g and
A2: f is A-measurable and
A3: g is A-measurable and
A4: f is without-infty and
A5: g is without-infty;
A6: dom(f+g) = dom f /\ dom g by A4,A5,Th16;
  f+g is A-measurable by A2,A3,A4,A5,Th31;
  then
A7: max-(f+g) is A-measurable by A1,A6,MESFUNC2:26;
A8: max-(f+g) is nonnegative by Lm1;
A9: max+f is nonnegative by Lm1;
  max+ f is A-measurable by A2,MESFUNC2:25;
  hence thesis by A7,A8,A9,Th31;
end;
