My textbook says the following when describing the conditions of a function to be a probability mass function on a random variable $\mathbb{x}$:
$\sum_{x \in \mathbb{x}} P(x) = 1$. We refer to this property as being normalized. Without this property, we could obtain probabilities greater than one by computing the probability of one of many events occurring.
Shouldn't it say "of many events occurring" instead of "of one of many events occurring"? The latter wouldn't necessarily be true, and it also seems out of context, since we're talking about the summation over all $x$ in the support?
Without normalization (as the author calls it), why do they have to be greater than one? Why couldn't it be the case that $\sum_x P(x) < 1$?
Either way, I wouldn't spend any time thinking about this too deeply. Any mass function $P \geq 0$ that sums to some positive number can be normalized by defining a new normalized probability mass function, call it $P^\prime$, by $P^\prime(x) = P(x) / \sum_y P(y)$.