# Approximate a set of functions by step functions on one partition

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 $\Omega_1,\ldots,\Omega_n$.

• Well, the wanted conclusion is a necessary and sufficient condition on $M$. It may help to clarify what you are looking for.
– js21
Aug 24, 2017 at 9:10

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
• Up to boundedness of $M$, you are correct. I completely overlooked that. What if we replace $\mathbb{R}$ by a separable Hilbert space $\mathcal{X}$? Aug 24, 2017 at 23:44