Let $X$ be a continuous real random variable, $Y \in \{0, 1\}$ be a discrete random variable and let $\mathcal D$ denote their joint probability distribution.
I've seen many times in the context of classification theory, describing the joint distribution $\mathcal D$ in terms of its $X$-marginal distribution $\mathcal D_X$ and the conditional probability $\eta(x) = \mathbb P[Y = 1 \mid X = x] = \mathbb E[Y \mid X = x]$.
My question regards whether the conditioning on $X=x$ in $\eta(x)$ poses constraints on the family of distributions $\mathcal D$ for which $\eta(x)$ is defined. Obviously, in the case where $\mathcal D_X$ has a pdf, $\eta(x)$ is well defined and can be expressed it terms of the pdf.
Question: For general distributions, where $X$ might not have a pdf, can $\eta(x) = \mathbb P[Y = 1 \mid X = x]$ still be defined?