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I am having problem understanding the idea of conditional expectation with respect to a sigma algebra. Is there any reference which explains the concept in detail. I will prefer something which does not go to geometric intuitions such as orthogonal projections.

HelloWorld
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  • Cinlar, "Probability and stochastics," covers everything from first principles. – sven svenson Sep 24 '20 at 15:45
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    I was only able to understand the notion of the conditional expectation with respect to a sub-$σ$-algebra $\mathcal G$, when I realized that this game is only interesting when $\mathcal G$ is "not Hausdorff", meaning that there might be points $x$ and $y$ which cannot be separated by a $\mathcal G$-measurable set. Any $\mathcal G$-measurable function must therefore coincide on $x$ and $y$, so $E(X|\mathcal G)$ tries to be the best photograph of the random variable $X$ which coincides on $x$ and $y$, as well as on any other similar pairs of points. – Ruy Sep 24 '20 at 20:27

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You can think of $E(X|\mathcal{G})$ as an approximation to a random variable $X$ that is $\mathcal{G}$-measurable. Eventually, it turns out to be the best possible $\mathcal{G}$-measurable approximation according to the least squares criterion. But you came here for references:

  1. The wikipedia page is fine to see the evolution of the concept, and it is necessary to fully understand the following references.
  2. Rick Durrett's book explains the modern definition in detail.
  3. This document clarifies the relationships between conditional expectation and conditional distributions, which, I think, is the core of the concept.
Álvaro G. Tenorio
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  • I am having trouble understanding the equation $$\mathbb{E}(Y\mid B)=\mathbb{E}(Y\mid \sigma (B))(B) $$. What I understand is that $\mathbb{E}(Y\mid \sigma(B))$ is a $\sigma(B)$ measurable function .But I dont understand how it can be evaluated at B instead of $\omega \in \Omega$ like other random variables. –  Sep 24 '20 at 15:53
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    @smbch $\sigma(B)={\emptyset,\Omega,B,\Omega\setminus B}$, and you can explicitly show that $E(Y|\sigma(B))$ is constant in $B$, so it is just an abuse of notation. $E(Y|B)$ is referring to the classical definition (see wikipedia) of conditional expectation with respect to an event (which is a number). – Álvaro G. Tenorio Sep 24 '20 at 15:59
  • @ Álvaro G. Tenorio Okay thanks.Will it be possible to give a reference which works this out in detail? –  Sep 24 '20 at 16:08
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    @smbch if you are refering to the proof that $E(Y|\sigma(B))$ is constant over $B$, it is fully detailed on Durrett's book (the begining of chapter $4$). – Álvaro G. Tenorio Sep 24 '20 at 16:13
  • @ Álvaro G. Tenorio I cant seem to find it. Can you please tell the page number? –  Sep 24 '20 at 16:32
  • @smbch It is a particular case of the example 4.1.5 (page 208) – Álvaro G. Tenorio Sep 24 '20 at 16:37