let X be non empty set ; :: thesis: for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,ExtREAL
for E being Element of S st dom (f . 0 ) = E & ( for n being natural number holds f . n is_measurable_on E ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is_measurable_on E
let S be SigmaField of X; :: thesis: for f being with_the_same_dom Functional_Sequence of X,ExtREAL
for E being Element of S st dom (f . 0 ) = E & ( for n being natural number holds f . n is_measurable_on E ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is_measurable_on E
let f be with_the_same_dom Functional_Sequence of X,ExtREAL ; :: thesis: for E being Element of S st dom (f . 0 ) = E & ( for n being natural number holds f . n is_measurable_on E ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is_measurable_on E
let E be Element of S; :: thesis: ( dom (f . 0 ) = E & ( for n being natural number holds f . n is_measurable_on E ) & ( for x being Element of X st x in E holds
f # x is convergent ) implies lim f is_measurable_on E )
assume A1: dom (f . 0 ) = E ; :: thesis: ( ex n being natural number st not f . n is_measurable_on E or ex x being Element of X st
( x in E & not f # x is convergent ) or lim f is_measurable_on E )
then A2: dom (lim_sup f) = E by Def9;
assume that
A3: for n being natural number holds f . n is_measurable_on E and
A4: for x being Element of X st x in E holds
f # x is convergent ; :: thesis: lim f is_measurable_on E
A5: dom (lim f) = E by A1, Def10;

A6: now

let x be Element of X; :: thesis: ( x in dom (lim f) implies (lim f) . x = (lim_sup f) . x )
assume A7: x in dom (lim f) ; :: thesis: (lim f) . x = (lim_sup f) . x
then f # x is convergent by A5, A4;
hence (lim f) . x = (lim_sup f) . x by A7, Th14; :: thesis: verum

end;

lim_sup f is_measurable_on E by A1, A3, Th23;
hence lim f is_measurable_on E by A5, A2, A6, PARTFUN1:34; :: thesis: verum