suppose I have a family of i.i.d standard normal random variables $Y_{n,k}$ and I define $X^N_t:=\sum_{n=0}^N\sum_{k=1}^{2^n}Y_{n,k}\phi_{n,k}(t)$ for $t\in [0,1]$ where $\phi_{n,k}$ are the Schauder functions.Furthermore, I know that $(X_t^N)$ is a martingale bounded in $L^2$ and therefore converges a.s. and in $L^2$ to a random variable $X$.
Why am I allowed to interchange expectation and the sum in the following expression
$E[X_sX_t]=\sum\sum E[Y_{n,k}Y_{l,m}]\phi_{n,k}\phi_{l,m}$
Note: The two sums are running both over two variables. The first over $n,m$ and the second over $k,l$.