Dear all, in order to prove the validity of my Galerkin approach of a certain variational problem, I need to check the so-called approximability property. In my case, it boils down to showing that for all $w\in L^2(\Omega)$, $\lim_{h\rightarrow 0}\inf_{w^h\in V^h}||w-w^h||=0$, where $\Omega=[0; 1]^d$, and $V^h$ is the space of piecewise constant functions on a regular (orthogonal) grid, with step $h$.

It is probably a classical result. However, I've browsed quickly the finite element literature, and the regularity requirements on $w$ are usually stronger ($H^1$ for example).

So, does the approximability property of $L^2$ functions by piecewise constant function hold? If yes, what theorem/author can I refer to?

Thanks in advance. Best regards, Sebastien