Consider a bounded sequence $f_n\in L^2(X,\mu)$, $\|f_n\|_{L^2}\leq C$.
Is the following true: if $f_n \to f$ weakly (that is $\langle f_n,g \rangle \to \langle f,g \rangle$ for every $g\in L^2(X,\mu)$), then Cesaro averages $\frac{1}{n} \sum_{k=1}^n f_k$ converge to $f$ almost everywhere?
What about other relations between convergence of a sequence and convergence of its Cesaro averages?
Added afterwards:
The first statement is false. Consider $f_n(x) = \sin kx$ for $10^{k-1}\leq n < 10^k$. Obviously, $f_n\to^w 0$, while $| \frac{1}{10^n} \sum_{k=1}^{10^n} f_k - \sin nx|\leq \frac{1}{10}$, meaning that $\limsup_n |\frac{1}{n} \sum_{k=1}^n f_k(x)| > 0$ for a.e. $x\in X$.
Question about other types of convergence remains open: Assuming that a sequence $f_n\in L^2$ converges weakly, can we say anything about the sequence of Cesaro averages $\frac{1}{n} \sum_{k=1}^n f_k$? Does it converge strongly, in measure?..
In fact, the question is motivated by the following problem: Fix $\varepsilon >0$. Is it true that $$ \mu(\{ x\in X : \limsup_n |\frac{1}{n} \sum_{k=1}^n f_k(x)| > \varepsilon \}) < \varepsilon $$ whenever $\|f_n\|_{L^2}< \delta$ for small enough $\delta = \delta(\varepsilon)$?
I will be grateful for any useful comments, suggestions or references!
The answer is negative.
If $(r_n)$ is lacunary sequences (that is $r_{n+1}>ar_n$ for some $a>1$), then for any probability preserving transformation $T\colon X\to X$ and any $\delta>0$ there is $A\subset X$ with $0<\mu(A)<\delta$ such that $$\limsup_n \frac{1}{n} \sum_{k=0}^{n-1} \chi_A (T^{r_k}x)=1 \quad \text{ for a.e. $x\in X$.}$$
Akcoglu M, Bellow A, Jones RL, Losert V, Reinhold–Larsson K, Wierdl M (1996) The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergodic Theory Dynam Systems 16(2):207–253.