# Alternative definitions of regular conditional distribution.

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.

• In the second kind do you still get a finite measure? If so can you not somehow use the characterisation via $\pi$-System by checking that $\mu(\cdot,\omega) = \tilde{\mu}(\cdot, \omega)$, this seems to give $\mathbb P$-a.s. the same regular conditional distribution, which seems to enough for almost everything, no? Commented May 19, 2023 at 14:53
• Clearly if we consider the first definition and the second, they agree almost surely. I believe the second is just obtained from the first, by just defining arbitrarily for all $\omega$ in the exceptional negligible set on which they differ, $\mu((-\infty, x),\omega)=F(x)$ for some cdf $F(x)$, which has nothing to do with $X$, and then using the $\pi-\lambda$ Theorem to have a well defined measure $\mu(\cdot, \omega)$ on this negligible set. The question is: why bother? What are the benefits of doing so? The subsequent theory seems to do just fine with the first approach.
– xyz
Commented May 20, 2023 at 12:43