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 $\Omega_1,\ldots,\Omega_n$.
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$\begingroup$ Well, the wanted conclusion is a necessary and sufficient condition on $M$. It may help to clarify what you are looking for. $\endgroup$ – js21 Aug 24 '17 at 9:10
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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
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$\begingroup$ Up to boundedness of $M$, you are correct. I completely overlooked that. What if we replace $\mathbb{R}$ by a separable Hilbert space $\mathcal{X}$? $\endgroup$ – user3095304 Aug 24 '17 at 23:44
