TITLE: Integer translates of a scaling function
QUESTION [8 upvotes]: I think this is asked as a standard exercise in books about wavelets (e.g. exercise 7.2 in Mallat's book), but I couldn't find a proof.  Let $\phi$ be a scaling function (see definition below).  I would like to learn why 
$$\sum_{k\in\mathbb Z} \phi(x-k) = 1 $$
almost everywhere.  
Definition. A sequence of subspaces $\{V_j: j\in \mathbb{Z}\}$ of $L^2(\mathbb R)$ is
called a multiresolution analysis  if it satisfies the following:

$V_j \subset V_{j+1}$
$\bigcap_{j}V_j = \{0\}$
$\overline{\bigcup_jV_j} = L^2(\mathbb R)$
$f(x)\in V_j$ if and only if $f(2x) \in V_{j+1}$
There exists a function $\phi \in V_0$ such that
$\{\phi(x-k)\}_{k\in\mathbb Z}$ is an orthogonal basis for $V_0$

The function $\phi$ here is called as a scaling function.

REPLY [2 votes]: I think you will find the proof for this in Mallat 1989, 'Multiresolution approximations and wavelet orthonormal bases of L^2'. Theorem 1 (in particular Equations (23), (36)) is what you are after. It is not trivial, longer to prove than I immediately thought. Perhaps there is a very fast proof but I can't think of it now.