let X be non empty set ; :: thesis: for S being SigmaField of X

for E being Element of S

for F being Functional_Sequence of X,REAL

for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let S be SigmaField of X; :: thesis: for E being Element of S

for F being Functional_Sequence of X,REAL

for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let E be Element of S; :: thesis: for F being Functional_Sequence of X,REAL

for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let F be Functional_Sequence of X,REAL; :: thesis: for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let m be Nat; :: thesis: ( ( for n being Nat holds F . n is E -measurable ) implies (Partial_Sums F) . m is E -measurable )

set PF = Partial_Sums F;

defpred S_{1}[ Nat] means (Partial_Sums F) . $1 is E -measurable ;

assume A1: for n being Nat holds F . n is E -measurable ; :: thesis: (Partial_Sums F) . m is E -measurable

A2: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

then A4: S_{1}[ 0 ]
by A1;

for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A4, A2);

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

for E being Element of S

for F being Functional_Sequence of X,REAL

for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let S be SigmaField of X; :: thesis: for E being Element of S

for F being Functional_Sequence of X,REAL

for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let E be Element of S; :: thesis: for F being Functional_Sequence of X,REAL

for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let F be Functional_Sequence of X,REAL; :: thesis: for m being Nat st ( for n being Nat holds F . n is E -measurable ) holds

(Partial_Sums F) . m is E -measurable

let m be Nat; :: thesis: ( ( for n being Nat holds F . n is E -measurable ) implies (Partial_Sums F) . m is E -measurable )

set PF = Partial_Sums F;

defpred S

assume A1: for n being Nat holds F . n is E -measurable ; :: thesis: (Partial_Sums F) . m is E -measurable

A2: for k being Nat st S

S

proof

(Partial_Sums F) . 0 = F . 0
by Def2;
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A3: S_{1}[k]
; :: thesis: S_{1}[k + 1]

F . (k + 1) is E -measurable by A1;

then ((Partial_Sums F) . k) + (F . (k + 1)) is E -measurable by A3, MESFUNC6:26;

hence S_{1}[k + 1]
by Def2; :: thesis: verum

end;assume A3: S

F . (k + 1) is E -measurable by A1;

then ((Partial_Sums F) . k) + (F . (k + 1)) is E -measurable by A3, MESFUNC6:26;

hence S

then A4: S

for k being Nat holds S

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