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,ExtREAL
for m being Nat st ( for n being Nat holds
( F . n is_measurable_on E & F . n is without-infty ) ) holds
(Partial_Sums F) . m is_measurable_on E
let S be SigmaField of X; :: thesis: for E being Element of S
for F being Functional_Sequence of X,ExtREAL
for m being Nat st ( for n being Nat holds
( F . n is_measurable_on E & F . n is without-infty ) ) holds
(Partial_Sums F) . m is_measurable_on E
let E be Element of S; :: thesis: for F being Functional_Sequence of X,ExtREAL
for m being Nat st ( for n being Nat holds
( F . n is_measurable_on E & F . n is without-infty ) ) holds
(Partial_Sums F) . m is_measurable_on E
let F be Functional_Sequence of X,ExtREAL ; :: thesis: for m being Nat st ( for n being Nat holds
( F . n is_measurable_on E & F . n is without-infty ) ) holds
(Partial_Sums F) . m is_measurable_on E
let m be Nat; :: thesis: ( ( for n being Nat holds
( F . n is_measurable_on E & F . n is without-infty ) ) implies (Partial_Sums F) . m is_measurable_on E )
set PF = Partial_Sums F;
defpred S₁[ Nat] means (Partial_Sums F) . $1 is_measurable_on E;
assume A1: for n being Nat holds
( F . n is_measurable_on E & F . n is without-infty ) ; :: thesis: (Partial_Sums F) . m is_measurable_on E
(Partial_Sums F) . 0 = F . 0 by Def0;
then C1: S₁[ 0 ] by A1;
C2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then C03: ( (Partial_Sums F) . k is_measurable_on E & (Partial_Sums F) . k is without-infty ) by A1, ADD1e;
( F . (k + 1) is_measurable_on E & F . (k + 1) is without-infty ) by A1;
then ((Partial_Sums F) . k) + (F . (k + 1)) is_measurable_on E by C03, MESFUNC5:37;
hence S₁[k + 1] by Def0; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(C1, C2);
hence (Partial_Sums F) . m is_measurable_on E ; :: thesis: verum