let X be non empty set ; :: thesis: for F being Functional_Sequence of X,ExtREAL
for S being SigmaField of X
for E being Element of S
for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is E -measurable & F . n is without+infty ) ) holds
(Partial_Sums F) . m is E -measurable
let F be Functional_Sequence of X,ExtREAL; :: thesis: for S being SigmaField of X
for E being Element of S
for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is E -measurable & F . n is without+infty ) ) holds
(Partial_Sums F) . m is E -measurable
let S be SigmaField of X; :: thesis: for E being Element of S
for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is E -measurable & F . n is without+infty ) ) holds
(Partial_Sums F) . m is E -measurable
let E be Element of S; :: thesis: for m being Nat st F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is E -measurable & F . n is without+infty ) ) holds
(Partial_Sums F) . m is E -measurable
let m be Nat; :: thesis: ( F is with_the_same_dom & E = dom (F . 0) & ( for n being Nat holds
( F . n is E -measurable & F . n is without+infty ) ) implies (Partial_Sums F) . m is E -measurable )
assume that
A1: F is with_the_same_dom and
A2: E = dom (F . 0) and
A3: for n being Nat holds
( F . n is E -measurable & F . n is without+infty ) ; :: thesis: (Partial_Sums F) . m is E -measurable

now :: thesis: for n being Nat holds
( (- F) . n is E -measurable & (- F) . n is without-infty )

let n be Nat; :: thesis: ( (- F) . n is E -measurable & (- F) . n is without-infty )
E = dom (F . n) by A1, A2, MESFUNC8:def 2;
then - (F . n) is E -measurable by A3, MEASUR11:63;
hence (- F) . n is E -measurable by Th37; :: thesis: (- F) . n is without-infty
F . n is without+infty by A3;
then - (F . n) is without-infty ;
hence (- F) . n is without-infty by Th37; :: thesis: verum

end;

then (Partial_Sums (- F)) . m is E -measurable by MESFUNC9:41;
then (- (Partial_Sums F)) . m is E -measurable by Th42;
then A5: - ((Partial_Sums F) . m) is E -measurable by Th37;
dom ((Partial_Sums F) . m) = E by A1, A2, A3, Th46, MESFUNC9:29;
then dom (- ((Partial_Sums F) . m)) = E by MESFUNC1:def 7;
then - (- ((Partial_Sums F) . m)) is E -measurable by A5, MEASUR11:63;
hence (Partial_Sums F) . m is E -measurable by DBLSEQ_3:2; :: thesis: verum