Definition of a certain discrete $L^p$ space?

Question

In the paper "Uniqueness results in an extension of Pauli's phase retrieval problem" by Philippe Jaming I'm currently trying to understand the author mentions the space $L^2(\mathbb{Z} / M \mathbb{Z})$ and only calls it the discrete setting. I was wondering if anyone can tell me the definition of this space. Neither it nor this $M$ is mentioned anywhere before in the paper.

Answer 1

In general, $L^p(X)$ is the space of measurable functions $f$ on $X$ such that the $p$-norm on $f$, $(\int_X |f|^p)^\frac{1}{p}$, is finite. Reading the introduction to the paper, it seems the author is comparing the space $L^2(\mathbb{R}^d)$ (of "square-integrable" functions on the real plane) against the space $L^2(\mathbb{Z}/M\mathbb{Z})$, of square-integrable functions on the integers modulo $M$ for some integer $M$ - we call $\mathbb{Z}/M\mathbb{Z}$ 'discrete' as it can be counted, or enumerated by integers.