Continous Versions of Regular Conditional Distributions

Let $X$ and $Y$ be random variables and assume $X$ has a density $f_X(x)$. An inversion theorem for the conditional expectation $E(Y\mid X = x)$ is derived which generalizes and simplifies that of Yeh. As an immediate corollary an almost-sure version of Bartlett's formula for the conditional characteristic function of $Y$ given $X = x$ is obtained. This result is applied to show the existence under regularity conditions of a version of the regular conditional distribution $P\{dy\mid X = x\}$ which is well defined for those values of $x$ such that $f_X(x) \neq 0$.