Let $X$ and $Y$ have joint density function $f(x,y)$. For $y\in\mathbb{R}$ and $A\in\mathcal{R}$, define $$\mu(y,A)=\dfrac{\int_A f(x,y)dx}{\int_{\mathbb{R}} f(x,y)dx},$$ if $\displaystyle\int_{\mathbb{R}}f(x,y)dx>0$ and $\mu(y,A)=0$ otherwise. Prove that $\mu$ is a probability kernel from $\mathbb{R}$ to $\mathbb{R}$ and that $X\mid Y \sim \mu(Y)$.
This was on a previous candidacy exam and I have no idea how to prove the result. I found out that it is a problem from Durrett's "Probability: Theory and Examples", specifically problem 5.1.13. Any help would be appreciated.