I'm reading a paper and trying to understand the definition of probability involving Borel sets.
I have a matrix $A \in \mathbb{R}^{m\times n}$, two vectors $s \in \mathbb{R}^{n}$ and $y \in \mathbb{R}^{m}$ such that $y=As$. Let $\delta(y_a-(As)_a)$ be the Dirac distribution on the hyperplane $y_a = (As)_a$ where $a \in \{1,\dots,m\}$. Denote by $$f_a(s_i) = \int_{}\prod_{j \neq i} f_j(s_i) \delta(y_a-(As)_a), a \in \{1,\dots,m\}$$
The authors mentioned that for any Borel set $S$, we can write $$f_a(S) = \mathbb{P}\left(y_a - \sum_{j \neq i} A_{aj} s_j \in A_{ai} S\right)$$ where $\mathbb{P}(\cdot)$ is the probability over the random vector $(s_1,\dots,s_{i-1},s_{i+1},\dots,s_N)^T$, which is distributrf according to the product measure $f_j(s_i), j \neq i$ and $A_{ai} S = \{A_{ai} x: x \in S\}$.
I fail to understand why the second equality holds. Do I need to look for which values of $x \in S$ such that $y_a-(As)_a = 0$?