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 V121() ) ) 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 V121() ) ) 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 V121() ) ) 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 V121() ) ) 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 V121() ) ) 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 V121() ) ; :: thesis: (Partial_Sums F) . m is E -measurable

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

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 V121() ) ) 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 V121() ) ) 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 V121() ) ) 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 V121() ) ) 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 V121() ) ) 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 V121() ) ; :: thesis: (Partial_Sums F) . m is E -measurable

now :: thesis: for n being Nat holds

( (- F) . n is E -measurable & (- F) . n is V120() )

then
(Partial_Sums (- F)) . m is E -measurable
by MESFUNC9:41;( (- F) . n is E -measurable & (- F) . n is V120() )

let n be Nat; :: thesis: ( (- F) . n is E -measurable & (- F) . n is V120() )

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 V120()

F . n is V121() by A3;

then - (F . n) is V120() ;

hence (- F) . n is V120() by Th37; :: thesis: verum

end;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 V120()

F . n is V121() by A3;

then - (F . n) is V120() ;

hence (- F) . n is V120() by Th37; :: thesis: verum

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