This is problem 2 of chapter 3 of Leo Breiman's book, Probability. Let $X$ and $Y$ by independent random variables defined on $\mathbb{R}$, and let $B \in \mathcal{B}$, where $\mathcal{B}$ is the class of (one-dimensional) Borel sets of $\mathbb{R}$. The problem has two parts:
For any $B \in \mathcal{B}$, $\mathbb{P}(X \in B - y)$ is a $\mathcal{B}$-measurable function of $y$. Note that $\mathbb{P}(X \in A) \equiv \mathbb{P}(\omega:X(\omega) \in A)$, where $\omega \in \Omega$, with $\Omega$ being the sample space. I am not sure what exactly Breiman means by $B - y$, but it appears to mean shifting all intervals in $B$ to the left by $y \in \mathbb{R}$. I believe I have a proof for this part, so this part is not what I am concerned about.
Show $\mathbb{P}(X + Y \in B) = \int \mathbb{P}(X \in B - y)\mathbb{P}(Y \in \text{d}y)$. I barely understand what exactly is meant by the notation (particularly, $\mathbb{P}(Y \in \text{d}y)$), so any illumination on even that part would be extremely helpful. Breiman does say that Fubini's theorem is needed to prove this result.
Help?