Alternative definitions of regular conditional distribution.

Question

Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$, Durrett (Probability: Theory and Examples, $\S 4.1.3$) defines the regular conditional distribution of a random variable $X$ given a sub-sigma-algebra $\mathcal{G}$ by requiring that the map $\mu(\cdot,\omega)$ is a probability measure on the state space only for almost every $\omega\in\Omega$ (that is, $\mathbb{P}$-a.s.); Billingsley (Probability and Measure, Theorem $33.3$) instead requires that it is so for every $\omega\in\Omega$. Is there any difference in these two approaches, or are they equivalent in terms of the theories that derive from them? Thanks for any explanation you can provide.