reserve X for set;

theorem
  for S being SigmaField of X, M being sigma_Measure of S, T being
N_Measure_fam of S st (for A being set st A in T holds A is measure_zero of M)
  holds union T is measure_zero of M
proof
  let S be SigmaField of X, M be sigma_Measure of S, T be N_Measure_fam of S;
  consider F being sequence of S such that
A1: T = rng F by Th12;
  set G = M*F;
  assume
A2: for A being set st A in T holds A is measure_zero of M;
A3: for r being Element of NAT st 0 <= r holds G.r = 0.
  proof
    let r be Element of NAT;
    F.r is measure_zero of M by A2,A1,FUNCT_2:4;
    then M.(F.r) = 0. by MEASURE1:def 7;
    hence thesis by FUNCT_2:15;
  end;
  G is nonnegative by MEASURE1:25;
  then SUM(G) = Ser(G).0 by A3,SUPINF_2:48;
  then SUM(G) = G.0 by SUPINF_2:def 11;
  then SUM(M*F) = 0. by A3;
  then
A4: M.(union T) <= 0. by A1,Th11;
  0. <= M.(union T) by MEASURE1:def 2;
  then M.(union T) = 0. by A4,XXREAL_0:1;
  hence thesis by MEASURE1:def 7;
end;
