
theorem Th21:
for X be non empty set, S be SigmaField of X, E be sequence of S,
 f be PartFunc of X,ExtREAL st
  for n be Nat holds f is (E.n)-measurable holds
 f is (Union E)-measurable
proof
    let X be non empty set, S be SigmaField of X, E be sequence of S,
     f be PartFunc of X,ExtREAL;
    assume A1: for n be Nat holds f is (E.n)-measurable;

    for r be Real holds (Union E) /\ less_dom(f,r) in S
    proof
     let r be Real;
     deffunc F(Element of NAT) = E.$1 /\ less_dom(f,r);
     consider E1 be sequence of bool X such that
A2:   for n be Element of NAT holds E1.n = F(n) from FUNCT_2:sch 4;

     for n be Element of NAT holds E1.n = less_dom(f,r) /\ E.n by A2; then
     (union rng E1) = less_dom(f,r) /\ (union rng E) by Th20; then
A3:  Union E1 = less_dom(f,r) /\ (union rng E) by CARD_3:def 4;

     reconsider E1 as SetSequence of X;
     for n be Element of NAT holds E1.n in S
     proof
      let n be Element of NAT;
A4:   f is (E.n)-measurable by A1;
      E1.n = E.n /\ less_dom(f,r) by A2;
      hence thesis by A4,MESFUNC1:def 16;
     end; then
     rng E1 c= S by FUNCT_2:114; then
     reconsider E1 as SetSequence of S by RELAT_1:def 19;

     Union E1 = Union E /\ less_dom(f,r) by A3,CARD_3:def 4;
     hence thesis by PROB_1:17;
    end;
    hence f is (Union E)-measurable by MESFUNC1:def 16;
end;
