From the question Approximation in $L^2$ by piecewise constant functions, I know I can approximate $L^2$ functions by piecewise constant functions. I wonder whether this claim also holds for stochastic processes.
Given a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$. Suppose there is a $\mathbb{R}$-valued $\mathbb{F}$-adapted stochastic process $X$, such that $\sup_{0\leq t \leq T} \mathbb{E} \vert X(t) \vert^2 < \infty$. Its piecewise constant approximation is defined by $X^h(t) = X(nh)$ for $t \in [nh, (n + 1)h)$, and $X^h(T) = X(T)$ with $h = \frac TN$ for some positive integer $N$. Do we have $X^h$ converge to $X$ in $L^2_{\mathbb{F}}(\Omega, \mathbb{R})$ as $h$ tends to $0$?