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

for M being sigma_Measure of S

for k being positive Real

for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for k being positive Real

for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let M be sigma_Measure of S; :: thesis: for k being positive Real

for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let k be positive Real; :: thesis: for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let F be Functional_Sequence of X,REAL; :: thesis: ( ( for m being Nat holds F . m in Lp_Functions (M,k) ) implies for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) )

assume A1: for m being Nat holds F . m in Lp_Functions (M,k) ; :: thesis: for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

defpred S_{1}[ Nat] means (Partial_Sums F) . $1 in Lp_Functions (M,k);

(Partial_Sums F) . 0 = F . 0 by MESFUN9C:def 2;

then A2: S_{1}[ 0 ]
by A1;

_{1}[j]
from NAT_1:sch 2(A2, A3);

hence for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) ; :: thesis: verum

for M being sigma_Measure of S

for k being positive Real

for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for k being positive Real

for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let M be sigma_Measure of S; :: thesis: for k being positive Real

for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let k be positive Real; :: thesis: for F being Functional_Sequence of X,REAL st ( for m being Nat holds F . m in Lp_Functions (M,k) ) holds

for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

let F be Functional_Sequence of X,REAL; :: thesis: ( ( for m being Nat holds F . m in Lp_Functions (M,k) ) implies for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) )

assume A1: for m being Nat holds F . m in Lp_Functions (M,k) ; :: thesis: for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k)

defpred S

(Partial_Sums F) . 0 = F . 0 by MESFUN9C:def 2;

then A2: S

A3: now :: thesis: for j being Nat st S_{1}[j] holds

S_{1}[j + 1]

for j being Nat holds SS

let j be Nat; :: thesis: ( S_{1}[j] implies S_{1}[j + 1] )

assume S_{1}[j]
; :: thesis: S_{1}[j + 1]

then A4: ( (Partial_Sums F) . j in Lp_Functions (M,k) & F . (j + 1) in Lp_Functions (M,k) ) by A1;

(Partial_Sums F) . (j + 1) = ((Partial_Sums F) . j) + (F . (j + 1)) by MESFUN9C:def 2;

hence S_{1}[j + 1]
by A4, Th25; :: thesis: verum

end;assume S

then A4: ( (Partial_Sums F) . j in Lp_Functions (M,k) & F . (j + 1) in Lp_Functions (M,k) ) by A1;

(Partial_Sums F) . (j + 1) = ((Partial_Sums F) . j) + (F . (j + 1)) by MESFUN9C:def 2;

hence S

hence for m being Nat holds (Partial_Sums F) . m in Lp_Functions (M,k) ; :: thesis: verum