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.


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