
theorem Th13:
for X be non empty set, S be SigmaField of X,
 A,B be Element of S, er be ExtReal holds
   chi(er,A,X) is B-measurable
proof
   let X be non empty set, S be SigmaField of X,
   A,B be Element of S, er be ExtReal;
a1:Xchi(A,X) is B-measurable by MEASUR10:32;
a2:dom Xchi(A,X) = X by FUNCT_2:def 1;
   per cases;
   suppose er = +infty;
    hence chi(er,A,X) is B-measurable by a1,Th2;
   end;
   suppose er = -infty; then
W:  chi(er,A,X) = -Xchi(A,X) by Th2;
    Xchi(A,X) is B-measurable by MEASUR10:32; then
    -Xchi(A,X) is B-measurable by a2,MEASUR11:63;
    hence chi(er,A,X) is B-measurable by W;
   end;
   suppose er <> +infty & er <> -infty; then
    er in REAL by XXREAL_0:14; then
    reconsider r = er as Real;
a3: chi(er,A,X) = r(#)chi(A,X) by Th1;
    dom chi(A,X) = X by FUNCT_3:def 3;
    hence chi(er,A,X) is B-measurable by a3,MESFUNC1:37,MESFUNC2:29;
   end;
end;
