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



Approximation in $L^2$ by piecewise constant functions
A few word about the setting of $L^2$ approximation scheme by simple functions. If $(\Omega,\mathcal{A},\mu)$ is a measure space and $\mathcal{B}$ is a finite sub-algebra of $\mathcal{A}$, generated by the partition $\mathcal{E}$, the class $L^2(\Omega,\mathcal{B},\mu)$ is a finite dimensional subspace of $L^2(\Omega,\mathcal{A},\mu)$ consisting of simple functions which are constant on each element of $\mathcal{E}$. The best approximation of $ w\in L^2(\Omega,\mathcal{A},\mu)$ is the simple function that takes the value $\frac{1}{\mu(E)}\int\_E vd\mu$ on each set $E\in\mathcal{E}$. Further, any sequence of finite algebras $(\mathcal{B\_n})\_{n\in \mathbb{N}}$ that generate $\mathcal{A}$ as a $\sigma$ algebra, produce this way an approximation of the identity (the corresponding orthogonal projectors converge strongly to the identity). 


In general, you may be interested in the notion of [conditional expectation relative to a $\sigma$-algebra](http://en.wikipedia.org/wiki/Conditional_expectation#Conditioning_relative_to_a_subalgebra) (the orthogonal projector). 


