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I'm reading a paper about lattice point counting, and I don't understand a step in one of their preliminary proofs.

Let $\chi(\ \cdot \ , \rho)$ denote the characteristic function of an open ball of radius $\rho$ in $\mathbb{R}^d$, let $\Gamma \subset \mathbb{R}^d$ be a lattice of full rank. Now define a torus $\mathbb{T}^d$ to be the quotient $\mathbb{R}^d / \Gamma$. Then:

$$\displaystyle \int_{\mathbb{T}^d} \sum_{m \in \Gamma} \chi(m - k, \rho) \ \mathrm{d}k = \int_{\mathbb{R}^d} \chi(k, \rho) \ \mathrm{d}k.$$

The expression on the right is just the volume of a $d$-sphere. But could someone explain how the author went from the left to the right? I suspect that there is an interchanging of sum and integral going on here. I notice that the integral varies over points not in $\Gamma$, while the sum varies over points that are in $\Gamma$, so I am thinking that the sum "fills in" all the points which the integral misses, which is why we end up with an integral over $\mathbb{R}^d$. But I don't know how to make this rigorous. Can anyone help?

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    What does $dk$ mean? The two spaces are different, so the integrals can't be with respect to the same measure.... (To be honest my intuition says that the sum over $m\in\Gamma$ should be on the $\Bbb R^d$ side—but let's find out first what the exact intended statement is.) – 2017-02-09
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    The aforementioned expression appears at the bottom of page 5, [here](http://www.ucl.ac.uk/~ucahnsi/Papers/mmnp-parnovsky.pdf). – 2017-02-09
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    I have no idea what the left-hand side is supposed to mean. :( $k$ is a point in $\Bbb T^d$, but $\chi$ is a function defined on $\Bbb R^d$. – 2017-02-09

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