reserve X for set;
reserve X,X1,X2 for non empty set;
reserve S for SigmaField of X;
reserve S1 for SigmaField of X1;
reserve S2 for SigmaField of X2;
reserve M for sigma_Measure of S;
reserve M1 for sigma_Measure of S1;
reserve M2 for sigma_Measure of S2;

theorem
for f being PartFunc of X,ExtREAL st f is_integrable_on M holds
  f is_a.e.integrable_on M
proof
   let f be PartFunc of X,ExtREAL;
   assume A1: f is_integrable_on M;
   reconsider A = {}, XX = X as Element of S by MEASURE1:7;
A2:M.A = 0 & A c= dom f & f|(dom f \ A) is_integrable_on M
     by A1,VALUED_0:def 19; then
   f|A` is_integrable_on M by Th15;
   hence thesis by A2;
end;
