3
$\begingroup$

When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about it is restricted to sets of a given sub-sigma-algebra, which is usually not finite. The article does not otherwise use probabilistic terminology, so it would be nice to not make it necessary for readers to be familiar with conditional expectations.

Does there exist established terminology or notation for the conditional expectation outside probabilistic context?

I will certainly mention the probabilistic interpretation. Otherwise, the paper uses classical epsilon-delta analysis, functional analysis, and a smattering of calculus of variations. The motivation is in PDE and inverse problems.

The function turns out to be in all $L^p$ spaces, and in particular in $L^\infty$.

$\endgroup$
  • $\begingroup$ What kind of function? Integrable? In $L_2$? What kind of sub-sigma-algebra --- finite perhaps? And how is the restricted information about the function presented? As a function? Or as a measure on the sub-sigma-algebra? $\endgroup$ – user95282 Jun 27 '18 at 17:29
  • $\begingroup$ @user95282 The function turns out to be in $L^\infty$. The sub-sigma-algebra is not finite. I edited the question to include these answers. $\endgroup$ – Tommi Jun 28 '18 at 8:42
3
$\begingroup$

I quote "Functional Analysis for Probability and Stochastic Processes: An Introduction" by Adam Bobrowski, introduction to Chapter 3 "Conditional expectation": $\newcommand{\cF}{\mathcal{F}}\newcommand{\cG}{\mathcal{G}}\newcommand{\PP}{\mathbb{P}}$

The space $L^2(\Omega,\cF,\PP)$ of square integrable random variables on a probabilty space $(\Omega,\cF,\PP)$ is a natural example of a Hilbert space. Moreover, for $X\in L^2(\Omega,\cF,\PP)$ and a $\sigma$-algebra $\cG\subset\cF$, the conditional expectation $\mathbb{E}(X|\cG)$ of $X$ is the projection of $X$ onto the subspace of $\cG$-measurable square integrable random variables.

The full definition (for random variables in $L^1(\Omega,\cF,\PP)$) appears a bit later in the book as Definition 3.2.5 and the conditional expectation in $L^p$ with $p>1$ is explained in Section 3.3.3.

I'd think that a reference to this book (or these chapters) would be very helpful for all readers who are not so familiar with probability and more familiar with functional analysis. So I'd suggest to use the word projection in this context (which has to understood with respect to the $L^2$ norm and not with respect the $L^\infty$ norm, but one can explain this in a few words, and Bobrowski in a good reference for any reader who wants to know more).

$\endgroup$
2
$\begingroup$

Yes, it's called a pushforward!

For more details, see this answer: Conditional Expectation for $\sigma$-finite measures

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.