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 $f_X(x) \neq 0$.
@article{1176995158,
author = {Zabell, Sandy},
title = {Continous Versions of Regular Conditional Distributions},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 159-165},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995158}
}
Zabell, Sandy. Continous Versions of Regular Conditional Distributions. Ann. Probab., Tome 7 (1979) no. 6, pp. 159-165. http://gdmltest.u-ga.fr/item/1176995158/