Assume that the two-dimensional random vector $X = (X_1 , X_2)$ has joint p.d.f. $f(x_1 , x_2 )$. That is, $f$ is a non-negative Borel function s.t. $\int fdy = 1$, and for any $x \in R^2 , P(X \leq x) = \int 1_{(−∞,x]} (y)f(y)dy$, where $dy$ denotes integration w.r.t. m^2 , two-dimensional Lebesgue measure.
(a) Prove that for any Borel set $B$ in $\mathfrak{B}(R^2 ), P(X \in B) = \int 1_B (y)f(y)dy$.
(b) Assume that $X$ is uniformly distributed on $[0, 2]^2$, that is,
$f(x) = \frac{1}{4}1_{[0,2]}(x_1)1_{[0,2]}(x_2)$.
Find the d.f. and p.d.f. of $Y = X_1 + X_2$.
I thought for a long time about (a) but did not make and progress. I tried but failed to express the set $B$ as some (preferably disjoint) union of the intervals of the form $(-\infty, x]$. Any hint is appreciated, thank you.
For (a), to show that $$\left\{B: P(X\in B)=\int_B f(x)\,dx\right\}$$ consists of all Borel sets, it suffices to show that it is a $\sigma$-algebra containing the sets of the form $(-\infty,x]$.