
theorem Th10:
  for X be non empty set, S be SigmaField of X, f be PartFunc of X,ExtREAL,
      A be Element of S st A c= dom f holds
    f is A-measurable iff
      max+f is A-measurable & max-f is A-measurable
proof
   let X be non empty set, S be SigmaField of X, f be PartFunc of X,ExtREAL,
   A be Element of S;
   assume A1: A c= dom f;
   hence f is A-measurable implies
    max+f is A-measurable & max-f is A-measurable by MESFUNC2:25,26;
   assume A2: max+f is A-measurable & max-f is A-measurable;
A3:dom (max-f) = dom f by MESFUNC2:def 3;
   now let r be Real;
    per cases;
    suppose r > 0; then
     less_dom(f,r) = less_dom(max+f,r) by Lm1;
     hence A /\ less_dom(f,r) in S by A2,MESFUNC1:def 16;
    end;
    suppose r <= 0; then
     less_dom(f,r) = great_dom(max-f,-r) by Lm1;
     hence A /\ less_dom(f,r) in S by A1,A2,A3,MESFUNC1:29;
    end;
   end;
   hence f is A-measurable by MESFUNC1:def 16;
end;
