How to describe the mapping from Random Variable $(X, Y)$ to a function of $X$? For example, $g(X):=E[Y|X]$.

Question

Suppose we have random variables $X\in\mathbb{R}^p$ and $Y\in\mathbb{R}$, the conditional expectation $\operatorname{E}[Y|X]:=\operatorname{E}[Y|\sigma(X)]$ is a function of $X$.

Define $g(X):=\operatorname{E}[Y|X]$, then $g(\cdot)$ is a function $\mathbb{R}^p\mapsto \mathbb{R}$

Such an operator (taking conditional expectation) maps the random variable $(X,Y)$ to a measurable function of $X$. I want to describe such operators in general, where the intention is to describe a class of predictors that can be derived from the joint distribution of $(X,Y)$. Is it a mapping from Function Space to Function Space? Is there a professional name for this kind of operators?

Thank you so much for any comments!

Answer 1

Does the following definition of conditional expectation suffice?

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathcal{G}$ be a $\sigma$-algebra contained in $\mathcal{F}$. For any real random variable $X \in L^1(\Omega, \mathcal{F}, P)$, define $E[X|\mathcal{G}]$ to be the unique random variable $Z \in L^1(\Omega, \mathcal{G}, P)$ such that for every bounded, $\mathcal{G}$-measurable random variable $Y$, $E[XY] = E[ZY]$.

See the notes here for this definition and more: https://galton.uchicago.edu/~lhttp://galton.uchicago.edu/~lalley/Courses/383/ConditionalExpectation.pdfalley/Courses/383/ConditionalExpectation.pdf

Answer 2

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $\mathcal{G} \subset \mathcal{F}$ be a $\sigma$-algebra. Conditional expectation $E(\cdot \mid \mathcal{G})$ is a projection mapping $L^1(\Omega, \mathcal{F}, P)$ onto $L^1(\Omega, \mathcal{G}, P)$.

Thus in your case, we can say $E(\cdot \mid \sigma(\cdot)) : L^1(\Omega, \mathcal{F}, P) \times M(\Omega, \mathbb{R}^p) \to \bigcup_{X \in M(\Omega, \mathbb{R}^p)}L^1(\Omega, \sigma(X), P)$. I'm not really sure there is a name for this kind of object. There is a result that the measurable functions of $X$ are precisely the functions measurable wrt $\sigma(X)$.