A limit definition of regular conditional probability

Question

I am working with a proof that would greatly benefit from a definition of conditional probability along the lines of the obscure unreferenced alternative definition found in Wikipedia. A Wikipedia user in the talk page said

Ridiculously, the "Alternate definition" in THIS article is an example of a NON-REGULAR conditional probability.

so some issues seem to arise in this definition. An equivalent question has appeared many years ago on MathSE but has, unsurprisingly, received no answers; therefore I appeal to the help of the MO community. In detail, I would want to know if, given real-valued random variables $X,Y$ and a Borel set $C$ $$\lim_{\varepsilon \to 0}\frac{P(\{Y \in C\}\cap \{X\in B_{\varepsilon}(x)\})}{P(X \in B_{\varepsilon}(x))}\stackrel{?}{=}P(Y \in C|X=x)$$ where $B_\varepsilon(x)=(x-\varepsilon,x+\varepsilon)$. In the case $P(X=x)>0$, the conclusion would follow by using $B_{1/n}(x)\supseteq B_{1/(n+1)}(x),\,\forall n$ and $B_{1/n}(x)\downarrow \{x\}$ but of course I am interested in the case where $\{X=x\}$ has measure zero. I have tried to search for a reference that hints at this problem, but to no avail. Thank you for any help.

Answer 1

Essentially yes. Only essentially, because the conditional probability on the right hand has to be defined in the first place.

The limit exists almost everywhere in the topology of weak convergence of measures and defines a regular condition distribution.

For a proof od a more general result see: P. Pfanzagl. "Conditional Distributions as Derivatives." Ann. Probab. 7 (6) 1046 - 1050, December, 1979. https://doi.org/10.1214/aop/1176994897.

Answer 2

Let me first reformulate your question. The conditional probability $$ f(x) = P(Y\in C|X=x) $$ is well-defined almost everywhere with respect to the marginal distribution $\mu$ of the random variable $X$ (since $C$ is fixed, there is no need to mention regular conditional probabilities for defining $f$). Then you are asking about the almost everywhere converges of the averages $$ \frac1{2\epsilon} \int_{B_\epsilon(x)} f(t) \,d\mu(t) $$ to $f(x)$ as $\epsilon\to 0$. If $\mu$ is the Lebesgue measure, then this is precisely the classical Lebesgue differentiation theorem.

It is also valid for arbitrary measures on the real line $\mathbb R$ (or even on $\mathbb R^d$ with $d>1$), see the references given in this discussion, which, therefore, provides an answer to your question.

One should be very careful with the Lebesgue differentiation theorem for more general setups as it is equivalent to the so-called Vitali property for the considered measure and the family of averaging sets, and (contrary to the "obscure" Wikipedia claim you mention) does not hold for arbitrary metric spaces (even if one considers differentiation with respect to metric balls), see Derivation and martingales by Hayes and Pauc, Theorem III.1.2. For instance, the quoted result of Pfanzagl refers to the differentiation with respect to a certain Vitaly system (which, indeed, does exist in any separable metric space) rather than with respect to metric balls (he mentions, however, that cubes do provide a Vitaly system in $\mathbb R^d$).