Atomless probability space and Random Variable with continuous distribution

Let $(\Omega, \mathcal{F}, P)$ a probability space. We say that a set $A\in\mathcal{F}$ is an atom if $P(A) > 0$ and for all $B \in \mathcal{F}$ such that $B \subset A$ if $P(B) < P(A)$ implies $P(B)=0$.

We call a space atomless if it does not have any atoms.

I am trying to find a reference or a proof for the following theorem:

A probability space $(\Omega, \mathcal{F}, P)$ is atomless if and only if it admits a random variable with continuous distribution.

Here a part of my question is answered but I can not prove the other part.

It would be really helpful to have a standard reference for this result.