let X be non empty set ; :: thesis:
let S be SigmaField of X; :: thesis:
let f be with_the_same_dom Functional_Sequence of X,REAL; :: thesis:
let g be PartFunc of X,ExtREAL; :: thesis:
let E be Element of S; :: thesis:
assume that
A1: dom (f . 0) = E and
A2: for n being Nat holds f . n is E -measurable and
A3: dom g = E and
A4: for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ; :: thesis:
A5: dom (lim f) = E by A1, MESFUNC8:def 9;

let x be Element of X; :: thesis:
assume A6: x in dom (lim f) ; :: thesis:
then x in E by A1, MESFUNC8:def 9;
then f # x is convergent by A4;
then lim (f # x) = lim (R_EAL (f # x)) by RINFSUP2:14;
then g . x = lim (R_EAL (f # x)) by A4, A5, A6;
hence g . x = (lim f) . x by A6, Th14; :: thesis:

end;

then A7: g = lim f by A3, A5, PARTFUN1:5;
for x being Element of X st x in E holds
f # x is convergent by A4;
hence g is E -measurable by A1, A2, A7, Th21; :: thesis: