Regular Conditional Distributions

Conditioning on an event of probability zero — like asking "what's the distribution of tomorrow's return given today's return is exactly 2.137%" — is mathematically delicate, because the usual formula for conditional probability (dividing by the probability of the event) divides by zero. For discrete variables this never bites, since every value has positive probability, but continuous variables hit it constantly, since any single exact value has probability zero.

The practical fix used throughout probability is to define the conditional distribution as a limit or via conditional expectation rather than direct division, and this generally works fine — except that the resulting object is only guaranteed to behave like a proper probability distribution for almost every value of the conditioning variable, not literally every single one. A regular conditional distribution is the stronger, cleaner object: a single family of distributions, one genuine probability distribution attached to each possible value $x$, chosen consistently so that averaging over $x$ recovers the right conditional expectations for every function you could ask about, with no leftover exceptional set where things silently misbehave. Its existence is not automatic — it needs a technical condition (working on a "nice" space, formally a Polish space) that holds for essentially every space encountered in finance and statistics, so in practice you rarely have to worry about the exceptional case.

Why it matters at all: constructions like Markov chains, filtrations, and simulation of conditional models all implicitly assume you can pick "the" distribution of the next state given the current one, for every current state, and stitch these together into a consistent whole. Regular conditional distributions are the theorem that guarantees that stitching is legitimate rather than a convenient fiction.