let X be non empty set ; :: thesis:
let S be SigmaField of X; :: thesis:
let M be sigma_Measure of S; :: thesis:
let E be Element of S; :: thesis:
let F be with_the_same_dom Functional_Sequence of X,ExtREAL; :: thesis:
let P be PartFunc of X,ExtREAL; :: thesis:
assume that
A1: E = dom (F . 0) and
A2: E = dom P and
A3: for n being Nat holds F . n is E -measurable and
A4: P is_integrable_on M and
A5: for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x and
A6: for x being Element of X st x in E holds
F # x is convergent ; :: thesis:
A7: E = dom (lim_sup F) by A1, MESFUNC8:def 8;
A8: for x being Element of X
for k being Nat st x in E holds
( - (P . x) <= (F # x) . k & (F # x) . k <= P . x )

proof

let x be Element of X; :: thesis:
let k be Nat; :: thesis:
assume A9: x in E ; :: thesis:
then x in dom (F . k) by A1, MESFUNC8:def 2;
then A10: x in dom |.(F . k).| by MESFUNC1:def 10;
|.(F . k).| . x <= P . x by A5, A9;
then |.((F . k) . x).| <= P . x by A10, MESFUNC1:def 10;
then ( - (P . x) <= (F . k) . x & (F . k) . x <= P . x ) by EXTREAL1:23;
hence ( - (P . x) <= (F # x) . k & (F # x) . k <= P . x ) by MESFUNC5:def 13; :: thesis:

end;

A11: now :: thesis:

let x be Element of X; :: thesis:
assume A12: x in dom (lim_sup F) ; :: thesis:
for k being Nat holds (superior_realsequence (F # x)) . k >= - (P . x)

proof

let k be Nat; :: thesis:
A13: (superior_realsequence (F # x)) . k >= (F # x) . k by RINFSUP2:8;
(F # x) . k >= - (P . x) by A7, A8, A12;
hence (superior_realsequence (F # x)) . k >= - (P . x) by A13, XXREAL_0:2; :: thesis:

end;

then lim_sup (F # x) >= - (P . x) by MESFUN10:4;
then A14: (lim_sup F) . x >= - (P . x) by A12, MESFUNC8:def 8;

now :: thesis:

let y be ExtReal; :: thesis:
assume y in rng (F # x) ; :: thesis:
then ex k being object st
( k in dom (F # x) & y = (F # x) . k ) by FUNCT_1:def 3;
hence P . x >= y by A7, A8, A12; :: thesis:

end;

then P . x is UpperBound of rng (F # x) by XXREAL_2:def 1;
then P . x >= sup (rng (F # x)) by XXREAL_2:def 3;
then P . x >= sup ((F # x) ^\ 0) by NAT_1:47;
then A15: P . x >= (superior_realsequence (F # x)) . 0 by RINFSUP2:27;
(superior_realsequence (F # x)) . 0 >= inf (superior_realsequence (F # x)) by RINFSUP2:23;
then P . x >= lim_sup (F # x) by A15, XXREAL_0:2;
then P . x >= (lim_sup F) . x by A12, MESFUNC8:def 8;
hence |.((lim_sup F) . x).| <= P . x by A14, EXTREAL1:23; :: thesis:

end;

A16: now :: thesis:

let x be Element of X; :: thesis:
assume A17: x in dom (lim F) ; :: thesis:
then x in E by A1, MESFUNC8:def 9;
then F # x is convergent by A6;
hence (lim F) . x = (lim_sup F) . x by A17, MESFUNC8:14; :: thesis:

end;

A18: lim_sup F is E -measurable by A1, A3, MESFUNC8:23;
E = dom (lim F) by A1, MESFUNC8:def 9;
then lim F = lim_sup F by A7, A16, PARTFUN1:5;
hence lim F is_integrable_on M by A2, A4, A7, A18, A11, MESFUNC5:102; :: thesis: