let X be non empty set ; :: thesis:
let F be with_the_same_dom Functional_Sequence of X,ExtREAL; :: thesis:
let S be SigmaField of X; :: thesis:
let M be sigma_Measure of S; :: thesis:
let E be Element of S; :: 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_measurable_on E 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 ; :: thesis:
A6: lim_inf F is_measurable_on E by A1, A3, MESFUNC8:24;

hereby :: thesis:

let n be Nat; :: thesis:
A7: F . n is_measurable_on E by A3;
A8: E = dom (F . n) by A1, MESFUNC8:def 2;

now

let x be Element of X; :: thesis:
assume A9: x in dom (F . n) ; :: thesis:
then A10: x in dom |.(F . n).| by MESFUNC1:def 10;
|.(F . n).| . x <= P . x by A5, A8, A9;
hence |.((F . n) . x).| <= P . x by A10, MESFUNC1:def 10; :: thesis:

end;

then F . n is_integrable_on M by A2, A4, A8, A7, MESFUNC5:108;
hence |.(F . n).| is_integrable_on M by A8, A7, MESFUNC5:106; :: thesis:

end;

A11: E = dom (lim_inf F) by A1, MESFUNC8:def 8;
A12: lim_sup F is_measurable_on E by A1, A3, MESFUNC8:23;
A13: 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 A14: x in E ; :: thesis:
then x in dom (F . k) by A1, MESFUNC8:def 2;
then A15: x in dom |.(F . k).| by MESFUNC1:def 10;
|.(F . k).| . x <= P . x by A5, A14;
then A16: |.((F . k) . x).| <= P . x by A15, MESFUNC1:def 10;
then A17: (F . k) . x <= P . x by EXTREAL2:60;
- (P . x) <= (F . k) . x by A16, EXTREAL2:60;
hence ( - (P . x) <= (F # x) . k & (F # x) . k <= P . x ) by A17, MESFUNC5:def 13; :: thesis:

end;

now

let x be Element of X; :: thesis:
assume A18: x in dom (lim_inf F) ; :: thesis:
then A19: x in E by A1, MESFUNC8:def 8;
for k being Nat holds (inferior_realsequence (F # x)) . k <= P . x

proof

let k be Nat; :: thesis:
reconsider k1 = k as Element of NAT by ORDINAL1:def 13;
A20: (inferior_realsequence (F # x)) . k1 <= (F # x) . k1 by RINFSUP2:8;
(F # x) . k <= P . x by A13, A19;
hence (inferior_realsequence (F # x)) . k <= P . x by A20, XXREAL_0:2; :: thesis:

end;

then lim_inf (F # x) <= P . x by Th5;
then A21: (lim_inf F) . x <= P . x by A18, MESFUNC8:def 8;

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 A13, A19; :: thesis:

end;

then - (P . x) is LowerBound of rng (F # x) by XXREAL_2:def 2;
then - (P . x) <= inf (F # x) by XXREAL_2:def 4;
then - (P . x) <= inf ((F # x) ^\ 0) by NAT_1:48;
then A22: - (P . x) <= (inferior_realsequence (F # x)) . 0 by RINFSUP2:27;
(inferior_realsequence (F # x)) . 0 <= sup (inferior_realsequence (F # x)) by RINFSUP2:23;
then - (P . x) <= lim_inf (F # x) by A22, XXREAL_0:2;
then - (P . x) <= (lim_inf F) . x by A18, MESFUNC8:def 8;
hence |.((lim_inf F) . x).| <= P . x by A21, EXTREAL2:60; :: thesis:

end;

then lim_inf F is_integrable_on M by A2, A4, A11, A6, MESFUNC5:108;
hence |.(lim_inf F).| is_integrable_on M by A11, A6, MESFUNC5:106; :: thesis:
A23: E = dom (lim_sup F) by A1, MESFUNC8:def 9;

now

let x be Element of X; :: thesis:
assume A24: 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:
reconsider k1 = k as Element of NAT by ORDINAL1:def 13;
A25: (superior_realsequence (F # x)) . k1 >= (F # x) . k1 by RINFSUP2:8;
(F # x) . k >= - (P . x) by A23, A13, A24;
hence (superior_realsequence (F # x)) . k >= - (P . x) by A25, XXREAL_0:2; :: thesis:

end;

then lim_sup (F # x) >= - (P . x) by Th4;
then A26: (lim_sup F) . x >= - (P . x) by A24, 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 A23, A13, A24; :: 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 A27: 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 A27, XXREAL_0:2;
then P . x >= (lim_sup F) . x by A24, MESFUNC8:def 9;
hence |.((lim_sup F) . x).| <= P . x by A26, EXTREAL2:60; :: thesis:

end;

then lim_sup F is_integrable_on M by A2, A4, A23, A12, MESFUNC5:108;
hence |.(lim_sup F).| is_integrable_on M by A23, A12, MESFUNC5:106; :: thesis: