let X be non empty set ; :: thesis: for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,REAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is E -measurable
let S be SigmaField of X; :: thesis: for f being with_the_same_dom Functional_Sequence of X,REAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is E -measurable
let f be with_the_same_dom Functional_Sequence of X,REAL; :: thesis: for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & ( for x being Element of X st x in E holds
f # x is convergent ) holds
lim f is E -measurable
let E be Element of S; :: thesis: ( dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & ( for x being Element of X st x in E holds
f # x is convergent ) implies lim f is E -measurable )
assume A1: dom (f . 0) = E ; :: thesis: ( ex n being Nat st not f . n is E -measurable or ex x being Element of X st
( x in E & not f # x is convergent ) or lim f is E -measurable )
then A2: dom (lim f) = E by MESFUNC8:def 9;
assume for n being Nat holds f . n is E -measurable ; :: thesis: ( ex x being Element of X st
( x in E & not f # x is convergent ) or lim f is E -measurable )
then A3: lim_sup f is E -measurable by A1, Th18;
assume A4: for x being Element of X st x in E holds
f # x is convergent ; :: thesis: lim f is E -measurable
A5: now :: thesis: for x being Element of X st x in dom (lim f) holds
(lim f) . x = (lim_sup f) . x
let x be Element of X; :: thesis: ( x in dom (lim f) implies (lim f) . x = (lim_sup f) . x )
assume A6: x in dom (lim f) ; :: thesis: (lim f) . x = (lim_sup f) . x
then f # x is convergent by A2, A4;
hence (lim f) . x = (lim_sup f) . x by A6, Th15; :: thesis: verum
end;
dom (lim_sup f) = E by A1, MESFUNC8:def 8;
hence lim f is E -measurable by A2, A3, A5, PARTFUN1:5; :: thesis: verum