Let $X$ be a set of $N>0$ elements (with the counting measure) and consider a family of measurable functions $f\_i:X\to [0,1]$, for $i\in \mathbb N$.


Any function $f\_i$ can be seen as a point in the hypercube $[0,1]^N$, therefore by using a mesh of sub-hypercubes (all of the same diameter) it is possible to find **finitely many** points $g\_j\in[0,1]^N$ such that: for any $i \in\mathbb N$ there exists a $j$ with the property that:
$$
0<g\_j-f\_i<\delta
$$


where $\delta$ is a fixed number depending on the mesh. Indeed it is enough to take the corners of the small hypercubes. Clearly $\delta$ can be made arbitrarily small by refining the mesh, i.e. increasing the number of $g\_j$.


In other words we can approximate the functions $f\_i$ with finitely many functions $g\_j$ and with arbitrary precision. The key assumption is that $X$ is a finite set.




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Now assume that $X$ is a measure space (of finite measure), do we have a similar approximation result for a family of measurable functions $f\_i:X\to [0,1]$? This problem seems very tricky but I wonder if there is some known result (even partial), in this direction. probably one needs to put some more properties on the $f\_i$



Approximating a family of measurable functions
$\newcommand\de\delta\newcommand\N{\mathbb N}$Such an approximation is impossible in such generality.


Indeed, let $f\_i:=(1+r\_i)/2$ for $i\in\mathbb N$, where $(r\_i)\_{i\in\mathbb N}$ is the [Rademacher system](https://en.wikipedia.org/wiki/Rademacher_system) (of real-valued functions on $X:=[0,1]$). Then the $f\_i$'s are measurable functions from $X$ to $[0,1]$ and $\|f\_i-f\_j\|=1$ for all distinct natural $i$ and $j$, where $\|\cdot\|$ is the uniform norm. So, for any $\de\in(0,1/2)$ there is no finite set $G$ of real-valued functions on $[0,1]$ such that for each $i\in\N$ there is some $g\_i\in G$ such that $\|f\_i-g\_i\|\le\de$ (because then, by the norm inequality, the $g\_i$'s would have to be pairwise distinct, and hence the set $G$ would have to be infinite).


