# Equivalent condition for a function of a random variable to be continuous almost everywhere

Suppose $$X$$ and $$Y$$ are random variables, and let $$P_Y$$, $$P_X$$, and $$P_{Y \mid X = x}$$ denote the probability distribution of $$Y$$, the probability distribution of $$X$$, and the conditional probability distribution of $$Y$$ given $$X = x$$. Let $$f$$ be a function whose domain is the support of the random variable $$Y$$. Let a.e. and a.s. denote abbreviations of almost everywhere and almost surely.

Question: Is $$f$$ is continuous at $$P_Y$$-a.e. $$y$$ if and only if it is continuous at $$P_{Y \mid X = x}$$-a.e. $$y$$ for $$P_X$$-a.e. $$x$$?

More generally, is it correct to say that some property $$V$$ of the random variable $$Y$$ holds $$P_Y$$-a.s. if and only if it holds $$P_{Y \mid X = x}$$-a.s. for $$P_X$$-a.e. $$x$$?

I can believe that if property $$V$$ holds $$P_{Y \mid X = x}$$-a.s. for $$P_X$$-a.e. $$x$$, then it is possibly holds $$P_Y$$-a.s. since this seems similar to the result that a countable union of measure zero sets is measure zero (I don't have a proof, though). I am not even sure that the converse is true.

• It is unclear to me what the domain of $f$ is. Jul 31, 2019 at 18:01
• @GiuseppeNegro Suppose the random variable $Y$ has support $\Pi \subset \mathbb{R}^d$ (we assume Euclidean space for simplicity). Then, we can assume that $f$ has domain $\Pi$. Jul 31, 2019 at 18:03

The core of your question does not seem to be related to the function $$f$$ at all. If I understand you correctly, your problem is solved by the continuous version of the law of total probability (compare this question), which says that
$$P_Y(B)=\int_{\Bbb R^d} P_{Y|X=x}(B) P_X(dx)$$
for every measurable set $$B\subset \Bbb R^d$$ (sticking to the euclidean case for simplicity). This integral is equal to $$1$$ if and only if the integrand is equal to $$1$$ for $$P_X$$-almost all $$x\in \Bbb R^d$$ (since the integrand is $$[0,1]$$-valued and $$P_X$$ is a probability measure). In other words, something indeed happens $$P_Y$$-almost surely if and only if it happens $$P_{Y|X=x}$$-almost surely for $$P_X$$-almost all $$x\in \Bbb R^d$$.
• Thanks, it looks like this link states a continuous version of the law of total probability. I agree that the function $f$ is irrelevant here, which means that the more general assertion is true under measurability assumptions. Jul 31, 2019 at 19:22