Is the conditional density of $X$ given $Y=y$ the density of $X$ with respect to some probability measure $P(\cdot \mid Y=y)$? Let $(\Omega,\Sigma,P)$ be a probability space and let $X:\Omega \to \mathbb R$ and $Y:\Omega \to \mathbb R$ be jointly continuous random variables with density functions $f_X$ and $f_Y$, respectively, and joint density function $f_{X,Y}$. The book Introduction to Probability by Bertsekas and Tsitsiklis defines 
$$
f_{X \mid Y}(x \mid y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}.
$$
Question: For a given $y \in \mathbb R$ such that $f_Y(y) &gt; 0$, does there exist a probability measure $Q$ such that the distribution of $X$ with respect to $Q$ has density function $x \mapsto f_{X\mid Y}(x \mid y)$?

In other words, does there exist a probability measure $Q$ such that
$$
Q(X \in S) = \int_S f_{X \mid Y}(x \mid y) \, dx
$$
for all measurable subsets $S$ of $\mathbb R$?

If such a measure $Q$ exists, I might call it $P(\cdot \mid Y = y)$ rather than $Q$. The book seems to sidestep and avoid the thorny issue of how to define $P(\cdot \mid Y = y)$, but it feels to me that $P(\cdot \mid Y = y)$ is a vital concept that is missing from the exposition.
I believe that you have precisely defined the probability measure that you are interested in. 

Intuitively, when we have the joint distribution of $X$ and $Y$, we can plot the contours of their joint density in $\mathbb{R}^2$. If we then fix $Y = y$, we get the conditional density $f_{X|Y}(x|y)$, which is precisely the density of $X$ on this "slice" of $\mathbb{R}^2$ where $Y = y$. That is to say, conditioning on $Y$, we are simply converting to a one-dimensional density, in a particular subspace of $\mathbb{R}^2$. Hence, any usual rules that would apply to a univariate density will apply here.
I believe that you have precisely defined the probability measure that you are interested in. 

Intuitively, when we have the joint distribution of $X$ and $Y$, we can plot the contours of their joint density in $\mathbb{R}^2$. If we then fix $Y = y$, we get the conditional density $f_{X|Y}(x|y)$, which is precisely the density of $X$ on this "slice" of $\mathbb{R}^2$ where $Y = y$. That is to say, conditioning on $Y$, we are simply converting to a one-dimensional density, in a particular subspace of $\mathbb{R}^2$. Hence, any usual rules that would apply to a univariate density will apply here.