Let $W=\{W_t\}_{t\in[0;T]}$ be a Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{X_t\}_{t\in[0;T]}$ be a read-valued, predictable and square-integrable stochastic process.
In my lecture notes, I read that then there exists a sequence $\{X^n_t\}_{t\in[0;T]}$ of stochastic processes where $$ X^n_t = \sum_{i=0}^{K^n-1} Y^{n,i} \cdot \textbf 1_{(t^{n,i};t^{n,i+1}]}(t) $$ for some $K^n$, $0 = t^{n,0} < t^{n,1} < \ldots < t^{n,K^n} = T$ and $Y^{n,i} \in \mathcal L^2(\Omega,\mathcal F_{t^{n,i}},\mathbb P;\mathbb R)$ such that $$ \lim_{n\to \infty} \mathbb E\bigg(\int_0^T (X_t - X^n_t)^2 \mathrm dt\bigg) = 0. $$
I am looking for a proper reference to this theorem. Multiple references are also welcome. Thank you!