
theorem Th32:
  for X being non empty set, S being SigmaField of X, A,B being Element of S
    holds Xchi(A,X) is B-measurable
proof
   let X be non empty set, S be SigmaField of X, A,B be Element of S;
A1:dom(Xchi(A,X)) = X by FUNCT_2:def 1;
   for r be Real holds B /\ great_eq_dom(Xchi(A,X),r) in S
   proof
    let r be Real;
    per cases;
    suppose r > 0; then
     B /\ great_eq_dom(Xchi(A,X), r) = B /\ A by Lm08;
     hence B /\ great_eq_dom(Xchi(A,X), r) in S;
    end;
    suppose r <= 0; then
     great_eq_dom(Xchi(A,X),r) = X by Lm09; then
     B /\ great_eq_dom(Xchi(A,X), r) = B by XBOOLE_1:28;
     hence B /\ great_eq_dom(Xchi(A,X), r) in S;
    end;
   end;
   hence thesis by A1,MESFUNC1:27;
end;
