This is a philosophical/pedagogical question. Ordinarily, if $(\Omega,\mathcal{A},\mu)$ is a probability space, then "drawing random point $X\in\Omega$" means that for each $A\in\mathcal{A}$, the probability of $X$ falling into $A$ is $\mu(A)$. Aside from making intuitive sense, this suggests a quasi-algorithmic sampling process: Finitely partition $\Omega$ into sets of small measure, and draw a partition member from the resulting finite distribution, according to weight under $\mu$.
Now suppose that $\mathcal{A}=2^\Omega$ and that $\mu$ admits non-singleton atoms. The existence of such "monsters" is independent of ZFC and, to take an extreme case, it might be that $\mu(A)\in\{0,1\}$ for all $A\subseteq\Omega$. In such a setting, (how) can one meaningfully speak of drawing a random point $X\in\Omega$ according to $\mu$?