Given a probability space $\Omega$ and a countable set $M$ of measurable functions $f\colon \Omega\to \mathbb{R}$, I am looking for conditions on $M$ such that the following holds: For all $\varepsilon>0$, there exists a measurable partition $\Omega\_1,\ldots,\Omega\_n$ of $\Omega$, such that
$$
\sup\_{f\in M}\inf\_{g\in S(\Omega\_1,\ldots,\Omega\_n)}\|f-g\|\_{L^2(\Omega)}\leq \varepsilon,
$$
where $S(\Omega\_1,\ldots,\Omega\_n)$ denotes the set of step functions (simple functions, piecewise constant functions, $\ldots$) on the partition $\Omega\_1,\ldots,\Omega\_n$.



Approximate a set of functions by step functions on one partition
Your condition states precisely that your set is relatively compact in $L^2$. There are many other chracterisations---perhaps most famously the Kolmogorov one which is easy to find with google


