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Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Furthermore, let $X, Y$ be two random variables from our probability space to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.

I have learned the definition of both $\mathbb{E}[X | \mathcal{G}]$ as well as $\mathbb{E}[X | Y] := E[X | \sigma(Y)]$, however, I don't know what the following means $$E[X | Y=y]$$ where $y \in \mathbb{R}$. Does this just mean that the random variable maps any $\omega \in \Omega$ to $y$? This does not really make sense to me because then the $\sigma(Y)$ would just be $\Omega$ right?

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    $\begingroup$ This usually denotes a measurable function $f$ such that $E[X|Y] = f(Y)$. (Under the usual conditions — in particular if $Y$ is real-valued — such a function always exists, and it is defined up to a set of zero measure with respect to the distribution of $Y$.) $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jan 8, 2021 at 14:37
  • $\begingroup$ Are you saying that the value $y \in \mathbb{R}$ is irrelevant? $\endgroup$
    – timudk
    Commented Jan 8, 2021 at 17:57
  • $\begingroup$ What I want to say is that $E[X|Y=y]$ defines a function of $y$ (or, more precisely, it is an expression that defines a function of the free variable $y$, up to equality almost everywhere with respect to the distribution of $Y$). For a fixed $y$, this is typically not well-defined. Note that in a similar way, one commonly writes e.g. $\sin x$ for a function of $x$ rather than the value for a particular $x$. Just a useful short-hand notation. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jan 8, 2021 at 20:12
  • $\begingroup$ Oh that helps a lot; one question though: why would one not write $E[X | Y](y)$ just like we write $\sin(x)$? $\endgroup$
    – timudk
    Commented Jan 9, 2021 at 14:42
  • $\begingroup$ Also, could you point me to a proof about the existence of such a function if $Y$ is real-valued? $\endgroup$
    – timudk
    Commented Jan 9, 2021 at 18:57

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