What does this sentence *the conditional random variable $x|y$ exists under $\pi$* mean?

Question

I'm reading this paper and don't understand this sentence (in Theorem 3.1 page 28):

Assume that the conditional random variable $x|y$ exists under $\pi$ with probability distribution denoted $\pi^y(dx)$.

What does the phrase "exists under $\pi$" mean exactly?

Thank you in advance for your explanation!