# Where does the random come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$?

For continuous random variables $$X, Y$$ the conditional expectation $$\mathbb{E}[X | Y]$$ is itself a random variable. I understood this in the sense that for a realisation of $$Y$$ we can say

$$\mathbb{E}[X | Y=y] = \int_{-\infty}^{\infty}xf_{X|Y}(x|y)dx = \frac{1}{f_Y(y)}\int_{-\infty}^{\infty}xf_{X,Y}(x,y)dx.$$

And so I imagine a random elementary event happening on $$\Omega$$ that gives a corresponding result to $$Y$$ and allows conditioning $$X$$ as above.

The signature would then be something like this

$$\mathbb{E}[X | Y]: \Omega \to \mathbb{R}, \quad \omega \mapsto \mathbb{E}[X | Y(\omega)]$$

The conditional expectation here is random because $$Y$$ is a random variable that gives an output to the random occurrences on the magical space $$\Omega$$.

However in more advanced courses and textbooks I studied on the matter the conditional expectation is often introduced via sub-$$\sigma$$-algebras and then defined by some characterisation like this:

Let $$X \in L_1(\Omega, \mathcal{A}, \mathbb{P})$$ and $$Y \in L_1(\Omega, \mathcal{F}, \mathbb{P})$$ where $$\mathcal{F}$$ is a sub-$$\sigma$$-algebra of $$\mathcal{A}$$. Then

$$Y = \mathbb{E}[X | \mathcal{F}] \quad \iff \quad \forall F \in \mathcal{F}: \mathbb{E}[\mathbb{1}_F X] = \mathbb{E}[\mathbb{1}_F Y].$$

These concepts supposedly coincide as $$\mathbb{E}[X | Y] = \mathbb{E}[X | \sigma(Y)]$$ and $$\mathbb{E}[X | \mathcal{F}]$$ is understood to be a random variable as well. But the interpretation gets lost for me.

The above definition of $$\mathbb{E}[X | Y]$$ is coherent with the common idea that $$\Omega$$ is the set of outcomes of some (real) random experiment. Selecting an outcome $$Y(\omega)$$ allows for the calculation of $$\mathbb{E}[X | Y=y]$$.

But for $$\mathbb{E}[X | \mathcal{F}]$$ it is not at all clear to me how this concept is "calculated". What is $$\mathbb{E}[X | \mathcal{F}](\omega)$$? Can you give a non-trivial (non discrete) example of a random experiment and a continuous random variable such that this expression is understandable and has meaning in the model of that random experiment?

• Just as $X$ is actually a function, $X:\,\omega\mapsto X(\omega)$, $\mathbb{E}[X | \mathcal{F}]$ is also a function (of $\omega$) that is furthermore $\mathcal{F}$-measurable and the closest to $X$. The solution does not have to be a constant function. See stats.stackexchange.com/a/230559/7224 Aug 11, 2022 at 11:54
• $\Omega$ is far from being some "magical space:" it has an extremely concrete meaning. It can represent all individuals in a population, for instance. Each individual $\omega\in\Omega$ has a definite numerical property $X(\omega).$ That's obviously not random. What is random is the process of selecting one individual from the population. The probability distribution $\mathbb P$ describes the relative chances of selection, thereby endowing $X$ with a distribution, etc.
– whuber
Aug 11, 2022 at 15:45
• @Xi'an I understand that it is also a function of $\omega$, but it is not at all clear to me what that law is. I.e. given an $\omega$, what do you do with it? Aug 12, 2022 at 8:03
• @whuber This is clear from basic probability theory. But don't we all have the implicit understanding that elements of $\Omega$ are picked at random? If you eliminate that intuition then all that is left is deterministic relations between deterministic sets and their measures. Sorry if "magical space" offended you, but I don't see how that hyperbole is relevant to the question. Aug 12, 2022 at 8:11
• I cannot discern a definite answerable question. What kind of responses are you hoping for?
– whuber
Aug 12, 2022 at 12:44