If $c$ is constant, what's $E[X|X=c]$ and why?

Question

From this thread, a new question arose.

Given $(\Omega, \mathscr{F}, P)$ and a constant $c$, what's $E[X|X=c]$ and why?

I believe they suggested that $E[X|X=c]=c.$

I have however that for $B\in \mathscr{F},$

$$E(X|B)=\frac{E(X1_B)}{P(B)}.$$

If the distribution function of $X$ is continuous, $P(B)=P({X=x})=0!$

Actually, not sure about this since $P$ isn't a distribution function, but a measure. This article states:

When $P(H)=0$ (for instance if $Y$ is a [[continuous random variable]] and H is the event $Y=y$, this is in general the case), the [[Borel–Kolmogorov paradox]] demonstrates the ambiguity of attempting to define the conditional probability knowing the event $H$. The above formula shows that this problem transposes to the conditional expectation. So instead one only defines the conditional expectation with respect to a σ-algebra or a random variable.

Answer 1

Short answer : It depends on how to define $\mathbb{E}[X | X = c]$.

Longer answer : As you cited, the conditional expectation is defined with respect to a $\sigma$-algebra not to an event. In fact, the conditional expectation is a class of random variables. Therefore the answer can depend on the choice of the element in the conditional expectation class.

If we define $\mathbb{E}[X | X = c]$ as $\mathbb{E}[X | \sigma(X)](\omega)$ for $\omega$ satisfying $X(\omega) = c$, and if we use the $X$ as a verion of $\mathbb{E}[X | \sigma(X)]$ then $\mathbb{E}[X | X = c] := \mathbb{E}[X | \sigma(X)](\omega) := X(\omega) := c$.

However, if the event $C := \{\omega : X(\omega) = c\}$ is a measure zero set then we can define $Z$ such that $Z(\omega) = X(\omega),~~\forall \omega \notin C$ and $Z(\omega) = c' \neq c,~~\forall \omega \in C$. Then, $Z$ is also a version of $\mathbb{E}[X | \sigma(X)]$. Therefore, in this case, $\mathbb{E}[X | X = c] := \mathbb{E}[X | \sigma(X)](\omega) := Z(\omega) := c' \neq c$.

This example shows that $\mathbb{E}[X | Y = c]$ is not well-defined in general. However, in many cases, there exists a canonical representer of a conditional expectation. In this case, we can define $\mathbb{E}[X | Y = c]$ with respect to the canonical representer.