# What is the intuition behind conditional expectation in a measure-theoretic treatment of probability?

What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?

You may assume I know:

• what a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ refers to
• probability without measure theory really well (i.e., discrete and continuous random variables)
• what the measure-theoretic definition of a random variable is
• that a Lebesgue integral has something to do with linear combinations of step functions, and intuitively, it involves partitioning the $y$-axis (as opposed to the Riemann integral, which partitions the $x$-axis).

I haven't had time to learn measure-theoretic probability lately due to graduate school creeping in as well as other commitments, and conditional expectation is often covered as one of the last topics in every measure-theoretic probability text I've seen.

I have seen notations such as $\mathbb{E}[X \mid \mathcal{F}]$, where I assume $\mathcal{F}$ is some sort of $\sigma$-algebra - but of course, this looks very different from, say, $\mathbb{E}[X \mid Y]$ from what I saw in my non-measure-theoretic treatment of probability, where $X$ and $Y$ are random variables.

I was also surprised to see that one book I have (Essentials of Probability Theory for Statisticians by Proschan and Shaw (2016)), if I recall correctly, explicity states that conditional expectation is defined as a conditional expectation, rather than the conditional expectation, which implies to me that there's more than one possible conditional expectation when given two pairs of random variables. (Unfortunately, I don't have the book on me right now, but I can update this post later).

The Wikipedia article is quite dense, and I see words such as "Radon-Nikodym" which I haven't learned yet, but I would at least like to get an idea of what the intuition of conditional expectation is in a measure-theoretic sense.

• In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf – WalterJ Aug 2 '18 at 12:11
• – BCLC Aug 2 '18 at 12:33
• The expression $\mathbb E[X\mid Y]$ is shorthand for $\mathbb E[X\mid \sigma(Y)]$, where $$\sigma(Y) = \{Y^{-1}(B) : B\in\mathcal B(\mathbb R)\}$$ is the $\sigma$-algebra generated by $Y$. – Math1000 Aug 2 '18 at 12:37
• Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf – Math1000 Aug 2 '18 at 12:41
• @WalterJ 's remark about the $L^2$ setting can be generalized to the general $L^1$ setting by considering $E[X \mid \mathcal{F}]$ to be the $L^1$ limit of $E[X I_{|X| \leq n} \mid \mathcal{F}]$ as $n \to \infty$, where now the approximants can be defined using Hilbert space methodology. Cf. math.stackexchange.com/questions/1130804/… – Ian Oct 17 '18 at 18:25