As part of my self-study, I am reading a text that uses the following notation to explain the convolution of Dirac measures $\delta_{a}$ and $\delta_{b}$:
$[\delta_{a} * \delta_{b}](B) = \delta_{b}(T_{-a}(B)) = \begin{cases} 1 & \text{if } b \in T_{-a}(B) \Longleftrightarrow a + b \in B = \delta_{a+b}(B)\\ 0 & \text{else } \end{cases}$
(Note that B is a Borel set).
Would someone be able to quickly explain the notation for me, particularly $T_{-a}(B)$ and why the double implication $b \in T_{-a}(B) \Longleftrightarrow a + b \in B$ holds?
More generally, I am trying to understand why the following formula holds:
$\delta_{x}*\delta_{y} = \delta_{x+y}$