
theorem Th3:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S st
 M is sigma_finite holds COM M is sigma_finite
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S;
    assume M is sigma_finite; then
    consider E be Set_Sequence of S such that
A1:  for n be Nat holds M.(E.n) < +infty and
A2:  Union E = X by MEASUR11:def 12;

    for n be Nat holds E.n in COM(S,M)
    proof
     let n be Nat;
     E.n is Element of COM(S,M) by MESFUN14:27;
     hence E.n in COM(S,M);
    end; then
    reconsider E1=E as Set_Sequence of COM(S,M) by MEASURE8:def 2;
    for n be Nat holds (COM M).(E1.n) < +infty
    proof
     let n be Nat;
     M.(E.n) < +infty by A1;
     hence (COM M).(E1.n) < +infty by MESFUN14:35;
    end;
    hence COM M is sigma_finite by A2,MEASUR11:def 12;
end;
