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For me it is pretty clear how the conditional expectation

$$ \mathbb{E}\left[X\left|\right.\mathcal{G}\right]\quad (1) $$

of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal{G}$ is defined. In particular, $(1)$ is the almost-unique $\mathcal{G}$-measurable random variable such that

$$ \mathbb{E}[(X-\mathbb{E}\left[X\left|\right.\mathcal{G}\right])^2] = \inf_{Z\text{ is }\mathcal{G}\text{-measurable}}\mathbb{E}[(X-Z)^2]. $$

This said, how the conditional expected value

$$ \mathbb{E}\left[X\left|\right.Y=a\right], $$ where $Y$ is another random variable and $a\in\mathbb{R}$, is formally defined?

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If you understood conditional expectation with respect to a sub-$\sigma$-algebra (actually your definition only work for $X\in L^2$, for $L^1$ you should use Radon-Nikodym or limiting argument), then the conditional expectation $\mathbb{E}[X\mid A]$ with respect to an event $A$ is simply as the value of the conditional expectation with respect to the sub-$\sigma$-algebra $\{\varnothing,A,A^c,\Omega\}]$ evaluated at points of $A$ (since $\mathbb{R}$ is Hausdorff every $a\in A$ give the same answer). Note that since $\mathbb{E}[X\mid\mathcal{G}]$ is only defined as an element of $L^1(\mathcal{G})$ (i.e. modulo almost sure equivalence) and not the set of all integrable $\mathscr{G}$-measurable random variable $\mathscr{L}^1(\mathcal{G})$, this is ill-defined for null event $A$, and is well-defined if $A$ is non-null. Conventionally, if $A$ is null we define $\mathbb{E}[X\mid A]=0$ (similar to how we construct $\mathbb{E}[X\mid\mathcal{G}]$) for definiteness). In particular, we have $$ \mathbb{E}[X\mid A]\mathbb{P}(A)=\mathbb{E}[X1_A]. $$ Note that $Y=a$ is an event $Y^{-1}(a)$, so this gives $\mathbb{E}[X\mid Y=a]$.

There are authors who insist on promoting nonsense such as $\displaystyle\mathbb{E}[X\mid Y=a]=\int_{\mathbb{R}} x\frac{f_{X,Y}(x,a)}{f_Y(a)}\,\mathrm{d}x$ even when $\mathbb{P}(Y=y)=0$. If you encounter it in any book, please do the good deed and burn the book along with the author.

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