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 Nat holds f . n is E -measurable ) & 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 E -measurable

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 Nat holds f . n is E -measurable ) & 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 E -measurable

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 Nat holds f . n is E -measurable ) & 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 E -measurable

let g be PartFunc of X,ExtREAL; :: thesis: for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & 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 E -measurable

let E be Element of S; :: thesis: ( dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & 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 E -measurable )

assume that

A1: dom (f . 0) = E and

A2: for n being Nat holds f . n is E -measurable 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 E -measurable

A5: dom (lim f) = E by A1, MESFUNC8:def 9;

for x being Element of X st x in E holds

f # x is convergent by A4;

hence g is E -measurable by A1, A2, A7, Th21; :: thesis: verum

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 Nat holds f . n is E -measurable ) & 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 E -measurable

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 Nat holds f . n is E -measurable ) & 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 E -measurable

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 Nat holds f . n is E -measurable ) & 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 E -measurable

let g be PartFunc of X,ExtREAL; :: thesis: for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & 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 E -measurable

let E be Element of S; :: thesis: ( dom (f . 0) = E & ( for n being Nat holds f . n is E -measurable ) & 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 E -measurable )

assume that

A1: dom (f . 0) = E and

A2: for n being Nat holds f . n is E -measurable 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 E -measurable

A5: dom (lim f) = E by A1, MESFUNC8:def 9;

now :: thesis: for x being Element of X st x in dom (lim f) holds

g . x = (lim f) . x

then A7:
g = lim f
by A3, A5, PARTFUN1:5;g . x = (lim f) . x

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 9;

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;assume A6: x in dom (lim f) ; :: thesis: g . x = (lim f) . x

then x in E by A1, MESFUNC8:def 9;

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

for x being Element of X st x in E holds

f # x is convergent by A4;

hence g is E -measurable by A1, A2, A7, Th21; :: thesis: verum