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 ) & E = dom P ) and
A2: for n being Nat holds F . n is_measurable_on E and
A3: P is_integrable_on M and
A4: for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x and
A5: for x being Element of X st x in E holds
F # x is convergent ; :: thesis:
C0: E = dom (lim_sup F) by A1, MESFUNC8:def 9;
C1: lim_sup F is_measurable_on E by A1, A2, MESFUNC8:23;
C2: 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 C3: x in E ; :: thesis:
then x in dom (F . k) by A1, MESFUNC8:def 2;
then C4: x in dom |.(F . k).| by MESFUNC1:def 10;
|.(F . k).| . x <= P . x by A4, C3;
then |.((F . k) . x).| <= P . x by C4, MESFUNC1:def 10;
then ( - (P . x) <= (F . k) . x & (F . k) . x <= P . x ) by EXTREAL2:60;
hence ( - (P . x) <= (F # x) . k & (F # x) . k <= P . x ) by MESFUNC5:def 13; :: thesis:

end;

C5: now

let x be Element of X; :: thesis:
assume D12: 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:
k is Element of NAT by ORDINAL1:def 13;
then D14: (superior_realsequence (F # x)) . k >= (F # x) . k by RINFSUP2:8;
(F # x) . k >= - (P . x) by C2, D12, C0;
hence (superior_realsequence (F # x)) . k >= - (P . x) by D14, XXREAL_0:2; :: thesis:

end;

then lim_sup (F # x) >= - (P . x) by MESFUN10:4;
then D15: (lim_sup F) . x >= - (P . x) by D12, MESFUNC8:def 9;

now

let y be ext-real number ; :: thesis:
assume y in rng (F # x) ; :: thesis:
then ex k being set st
( k in dom (F # x) & y = (F # x) . k ) by FUNCT_1:def 5;
hence P . x >= y by C2, D12, C0; :: 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_1:48;
then D17: 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 D17, XXREAL_0:2;
then P . x >= (lim_sup F) . x by D12, MESFUNC8:def 9;
hence |.((lim_sup F) . x).| <= P . x by D15, EXTREAL2:60; :: thesis:

end;

C6: E = dom (lim F) by A1, MESFUNC8:def 10;

now

let x be Element of X; :: thesis:
assume P01: x in dom (lim F) ; :: thesis:
then x in E by A1, MESFUNC8:def 10;
then F # x is convergent by A5;
hence (lim F) . x = (lim_sup F) . x by P01, MESFUNC8:14; :: thesis:

end;

then lim F = lim_sup F by C0, C6, PARTFUN1:34;
hence lim F is_integrable_on M by C5, C0, C1, A1, A3, MESFUNC5:108; :: thesis: