Pointwise defined version of conditional expectation with respect to a random variable

TL;DR

This paper introduces a novel approach to defining conditional expectations pointwise with respect to a random variable, addressing limitations of traditional measure-theoretic methods and enabling more precise conditioning on singular events.

Contribution

It proposes a new method using the Lebesgue-Besicovich lemma to obtain pointwise conditional expectations without extra topological assumptions.

Findings

Provides a pointwise definition of conditional expectation

Addresses the gap between measure-theoretic objects and practical computation

Enables evaluation of conditional expectations at specific points

Abstract

It is often of interest to condition on a singular event given by a random variable, e.g. $\{Y=y\}$ for a continuous random variable $Y$. Conditional measures with respect to this event are usually derived as a special case of the conditional expectation with respect to the random variables generating sigma algebra. The existence of the latter is usually proven via a non-constructive measure-theoretic argument which yields an only almost-everywhere defined quantity. In particular, the quantity $\mathbb E[f|Y]$ is initially only defined almost everywhere and conditioning on $Y=y$ corresponds to evaluating $\mathbb E[f|Y=y] = \mathbb E[f|Y]{Y=y}$, which is not meaningful because of $\mathbb E[f|Y]$ not being well-defined on such singular sets. This problem is not addressed by the introduction of regular conditional distributions, either. On the other hand it can be shown that the naively computed conditional density $f_{Z|Y=y}(z)$ (which is given by the ratio of joint and marginal densities) is a version of the conditional distribution, i.e. $\mathbb E[\{Z\in B\}|Y=y] = \int_B f_{Z|Y=y}(z) dz$ and this density can indeed be evaluated pointwise in $y$. This mismatch between mathematical theory (which generates an object which cannot produce what we need from it) and practical computation via the conditional density is an unfortunate fact. Furthermore, the classical approach does not allow a pointwise definition of conditional expectations of the form $\mathbb E[f|Y=y]$, only of conditional distributions $\mathbb E[\{Z\in B\}|Y=y]$. We propose a (as far as the author is aware) little known approach to obtaining a pointwise defined version of conditional expectation by use of the Lebesgue-Besicovich lemma without the need of additional topological arguments which are necessary in the usual derivation.