Assume that there is a constant c such that for every nonnegative measurable function f, satisfying Ef(X)                  lim sup n→∞ E  n i=1 |Wn,i(X)|f(Xi)  ≤ cEf(X). Prove that (i) in Theorem 4.1...

Assume that there is a constant c such that for every nonnegative measurable function f, satisfying Ef(X)

                 lim sup n→∞ E  n i=1 |Wn,i(X)|f(Xi)  ≤ cEf(X).

Prove that (i) in Theorem 4.1 is satisfied (Proposition 7 in Stone (1977)). Hint: Apply an indirect proof.