Recall what we said about continuous mathematics: it studies structures analagous to $\mathbb{R}$, the set of real numbers. We said one of the defining features of $\mathbb{R}$ is the inability to neatly list out the numbers in $\mathbb{R}$ (this property is formally called “uncountability”). This makes some of our characterizations of random variables in the previous section tricky to generalize. For example, we said one of the features of pmfs was that

$$\sum_{x\in\mathbb{Z}}f_X(x)=1$$

You can think of this sum as going through the list of integers $\{\dots,-2,-1,0,1,2,\dots\}$. But as we said, we can’t make a list of the numbers in $\mathbb{R}$, so how would we sum over all $x\in\mathbb{R}$? Even if we were to only consider distributions with a bounded support, like the support of the binomial distribution ($\{0,1,\dots,n-1,n\}$), we can’t really talk about a listing of all real numbers between zero and $n$. So we need to make adjustments to our definitions. This first section will demonstrate how we can make these adjustments.