Let $(\Omega, F, \mathbb{P})$ be a probability space and let $(S, \Sigma)$ be a measurable space. Consider a random variable $X: \Omega \rightarrow S$. We say that $X$ is discrete if its range $X(\Omega)$ is finite or countably infinite.
But what if we say that $X$ follows some known discrete distribution without specifying the codomain $S$? For example, suppose that $X$ is Poisson-distributed. Then the range of $X$ is the set of nonnegative integers. But what is its codomain $S$? Is it the set of nonnegative integers, the set of integers, or the set of real numbers? And what is the sigma algebra $\Sigma$? The same question could be asked e.g. if $X$ follows a binomial distribution.