$\mathbb{R}^3$ denotes the set of all $3$-dimensional real vectors, and $[0,1]^3$ denotes the set of all $3$-dimensional real vectors each of whose components is between $0$ and $1$, inclusive.

The definition of $X$, namely $\{x \in \mathbb{R}^3 \mid A^N x \in [0,1]^3\}$, denotes the set of all elements $x$ of $\mathbb{R}^3$ satisfying $A^N x \in [0,1]^3$.

Unless otherwise stated, the volume of a subset of $\mathbb{R}^3$ is formulated using the Lebesgue measure. Specifically, the volume of a subset $U$ of $\mathbb{R}^3$ is defined only when $U$ is Lebesgue measurable, and its value is determined as the value obtained by substituting $U$ into the Lebesgue measure of $\mathbb{R}^3$ (or, equivalently, the value of the Lebesgue integral of the constant function $1$ over $U$).

For example, the volume of a rectangular cuboid whose $3$ edges extending from $1$ vertex have lengths $a,b,c$, respectively, is $abc$; in this way, the formulation is consistent with the volume formulas learned up through high school.

Since the subset $X$ considered here is known to be Lebesgue measurable, its volume $V$ is determined as a non-negative real number or $\infty$, and when it is a positive real number (that is, neither $0$ nor $\infty$), it can be proved that $V^{-1}$ turns out to be a positive integer.

Note that the $\infty$ appearing in integrals and the like is formulated by arbitrarily fixing one term known not to belong to $\mathbb{R}$ and calling it $\infty$. Since the arguments are made so that it does not matter which term is called $\infty$, how exactly it is chosen is often left unstated; but if a concrete formulation is needed, one can, for example, define $\infty$ to be $\mathbb{R}$ itself, and then it follows that $\infty$ does not belong to $\mathbb{R}$, that is, it can be distinguished from real numbers. Then, if necessary, an order and operations extending those of $\mathbb{R}$ are often defined as appropriate on $\mathbb{R} \cup \{\infty\}$ and used.