
theorem
  for X being non empty set, S being SigmaField of X, A being Element of S,
      p being R_eal holds
    X --> p is A-measurable
proof
   let X be non empty set, S be SigmaField of X, A be Element of S,
       p be R_eal;
A0:dom(X --> p) = X by FUNCOP_1:13;
   for r be Real holds A /\ great_eq_dom(X --> p, r) in S
   proof
    let r be Real;
    per cases;
    suppose A2: r > p;
     for x be object holds not x in great_eq_dom(X --> p,r)
     proof
      let x be object;
      hereby assume x in great_eq_dom(X-->p, r); then
       x in dom(X-->p) & r <= (X-->p).x by MESFUNC1:def 14;
       hence contradiction by A2,FUNCOP_1:7;
      end;
     end; then
     great_eq_dom(X --> p, r) = {} by XBOOLE_0:def 1;
     hence A /\ great_eq_dom(X --> p, r) in S by MEASURE1:7;
    end;
    suppose A4: r <= p;
     now let x be object;
      assume A6: x in X; then
      (X-->p).x = p by FUNCOP_1:7;
      hence x in great_eq_dom(X-->p,r) by A0,A4,A6,MESFUNC1:def 14;
     end; then
     X c= great_eq_dom(X-->p,r); then
     great_eq_dom(X --> p, r) = X; then
     A /\ great_eq_dom(X --> p, r) = A by XBOOLE_1:28;
     hence A /\ great_eq_dom(X --> p, r) in S;
    end;
   end;
   hence thesis by A0,MESFUNC1:27;
end;
