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 Th16:
for f being PartFunc of X,ExtREAL
 st f is_a.e.integrable_on M holds dom f in S
proof
    let f be PartFunc of X,ExtREAL;
    assume f is_a.e.integrable_on M; then
    consider A be Element of S such that
A1:  M.A = 0 & A c= dom f & f|A` is_integrable_on M;
    consider B be Element of S such that
A2:  B = dom(f|A`) & f|A` is B-measurable by A1,MESFUNC5:def 17;
    dom(f|A`) = dom f /\ A` by RELAT_1:61
     .= dom f /\ (X \ A) by SUBSET_1:def 4
     .= (dom f /\ X) \ A by XBOOLE_1:49
     .= dom f \ A by XBOOLE_1:28; then
    dom f = A \/ B by A1,A2,XBOOLE_1:45;
    hence dom f in S;
end;
