I recently witnessed statisticians discussing $[Y|X=x]$ as if it were a random variable. In particular, they were making assumptions about its distribution.
I am familiar with various definitions of conditional expectations, in particular with $\mathbb E[Y|X=x]$. And I know that $\mathbb E[Y|X]$ is a random variable. But don't remember ever seeing $[Y|X=x]$ as a random variable. What could be its definition?
A bit of context added: In the context of inference, let $X$ be the predictor r.v. and $Y$ the response. Define $r(x)= \mathbb E(Y|X = x)$ and call it the regression function. Then (according to the statisticians) we make an assumption on the distribution of $(Y|X = x)$.
Formally, $[Y|X=x]$ does not make sense. However, under some mild assumptions the probability measure $P(\,\cdot\,|X=x)$ exists for each $x$, so it makes sense to discuss the distribution of $Y$ with respect to this measure. This is what is really meant, and it is not surprising that a statistician would take this shortcut. After all, if you don't know any measure theory then what I have said might not mean anything to you, whereas talking about "conditional distributions" should have some intuition behind it - for example, if $Z$ is independent of $X$, the conditional distribution of $X^2-3XZ$ given $X=x$ should clearly be the same as the distribution of $x^2-3xZ$.