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 g being PartFunc 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 ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) holds
g is_measurable_on E
let S be SigmaField of X; :: thesis: for f being with_the_same_dom Functional_Sequence of X,REAL
for g being PartFunc 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 ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) holds
g is_measurable_on E
let f be with_the_same_dom Functional_Sequence of X,REAL ; :: thesis: for g being PartFunc 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 ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) holds
g is_measurable_on E
let g be PartFunc 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 ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) holds
g 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 ) & dom g = E & ( for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ) implies g is_measurable_on E )
assume that
A1: dom (f . 0 ) = E and
A2: for n being natural number holds f . n is_measurable_on E and
A3: dom g = E and
A4: for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ; :: thesis: g is_measurable_on E
A5: dom (lim f) = E by A1, MESFUNC8:def 10;

now

let x be Element of X; :: thesis: ( x in dom (lim f) implies g . x = (lim f) . x )
assume A6: x in dom (lim f) ; :: thesis: g . x = (lim f) . x
then x in E by A1, MESFUNC8:def 10;
then f # x is convergent by A4;
then lim (f # x) = lim (R_EAL (f # x)) by RINFSUP2:14;
then g . x = lim (R_EAL (f # x)) by A4, A5, A6;
hence g . x = (lim f) . x by A6, Th14; :: thesis: verum

end;

then A7: g = lim f by A3, A5, PARTFUN1:34;
for x being Element of X st x in E holds
f # x is convergent by A4;
hence g is_measurable_on E by A1, A2, A7, Th21; :: thesis: verum