What is the definition of $X|(Y=y)$? Suppose $S$ is a sample space (the set of all outcomes $\omega_i$) for an experiment. A random variable $X$ is defined as a real-valued function which maps elements from the sample space to real numbers, i.e. $X:S\to \mathbb R$.
Discrete Random variable:
The definition of the conditional probability mass function of $X$ given $Y=y$ is $$\mathbb P(X=x|Y=y)=\frac{\mathbb P(X=x, Y=y)}{\mathbb{P}(Y=y)} .$$
Question: In lecture slides I have seen the notation, for example, that $X|(Y=y) \sim \text{Bin}(m, \lambda).$
What is the definition of $X|(Y=y)$? Is it a random variable itself with a restricted sample space? Maybe $X|(Y=y): \{\omega\in S: Y(\omega)=y \} \to \mathbb R$?
What would be the definition of $X|(Y=y)$ for $X$ and $Y$ being continuous random variables?
(Note: If it isn't a random variable, then how can we talk about it's distribution and expected value?)
 A: Summarising the very helpful comments from @Nap D. Lover and @d.k.o. 
 - In the original theory of conditional probability, there is no such definition of a "conditional random variable."
Before addressing the notation, a thought about the "requirement" of a conditional random variable


*

*The purpose of a conditional distribution, $\mathbb P(X=x|Y=y)$, is a way to "recalibrate" the probability assignment/distribution for $X$, given we received information about $Y$. (Which intuitively, could be the probability distribution of the temperature $X$ as $\mathbb P(X=x)$ vs. the probability distribution of the temperature $X$, given the humidity $Y$ was $y$, being $\mathbb P(X=x|Y=y)$). It is still a probability distribution designed for the random variable $X$, just "recalibrated" to better model the "true" probabilities for the given situation.

*So I guess, in a way, a new random variable for a "conditional random variable" is not really necessary. While it is possible to define a random variable $X_y$ living on a new restricted sample space, maybe it moves away from the idea of this distribution being "rediagnosis" of what the probability distribution of $X$ should be, given the new "symptoms" ($Y=y$).

*Hence it makes sense to only need Conditional distributions and Conditional expectation (The expected value of $X$, but weighted in a different way to account for the new information) etc, and not a new random variable itself.
The notation:
So the interpretation of the notation can be left as what @d.k.o. said in the very first comment, $X|(Y=y) \sim \text{Bin}(m, \lambda)$ is just shorthand notation for saying "The distribution of $X$, conditioned on $Y=y$, is (from the definition in the question) $\text{Bin}(m, \lambda)$.
