Proof to motivate the need for measure theoretic probability

Let $x$ be an element of the probability space $X$. Let $p$ be the probability of the event $\{x\}$. Consider the $2^n$ pairwise disjoint sets of the form $A_1^{\varepsilon_1}\cap\dots\cap A_n^{\varepsilon_n}$ where for each $i\in\{1,\dots,n\}$, $\varepsilon_i\in\{0,1\}$, $A_i^0=A$ and $A^1_i=X\setminus A$. Each of these sets has probability $2^{-n}$ and $x$ is an element of one of them. It follows that $p\leq 2^{-n}$. Since this holds for every $n$, $p=0$. So every singleton in your space has probability $0$ and hence the space is not discrete.