How to formally define a conditional distribution conditioning on an event of probability zero? Given $[X,Y]\sim N(0,I_2)$, a intuitive guess of the value of $P(X=x|\{X,Y\}=\{x,y\})$, where $\{\}$ means unordered set,  is $1/2$ by symmetry. This type of notations is typically applied in conformal inference with assumption about exchangeability (see Conformal Prediction Under Covariate Shift).

For things like $P(X|Y=y)$, we know the regular conditional probability
(see Regular conditional probability in Wiki), and the conditions required for the latter one can be find in many statistic textbooks (e.g. Mathematical Statistics by Shao jun).
However, $set\{\}$ cannot be simply treated as a random variable or a measurable function about random variables, so it obscures the $\sigma$-algebra conditioned on. The question is how to regularize the underlying probability space in order to construct the respective regular conditional distribution.

A formal definition can be done by the theory of measurable correspondence. Situation here is trivial since we only need a $\sigma$-algebra with measure. Intuitively, here we can fold the $R^2$ along the $y=x$ and double the measure.
