Let $\{X(t) : t \in T\}$ be a separable stochastic process where $(T,d)$ is a metric space. Let $B(t,r) = \{u \in T : d(t,u)\leq r\}$ and, $\pi$ be a sigma-finite measure on $T$.
For $k\geq 1$, define the local smoothing operator $\mathbb{S}_k$
$$\mathbb{S}_k X (t) = \frac{1}{\pi(B(t,2^{-k}))} \int_{B(t,2^{-k})} X(u)~\pi(du)$$
I was wondering if $\lim_k \mathbb{S}_k X(t) = X(t)$ hold, and how can one show it.