I'm doing an exercise in which I need to apply Fubini–Tonelli theorem on a non-negative function $f$. So I have to check that $f$ is measurable. However, it turns out that applying the definition of Borel $\sigma$-algebra to a specific example is not as easy as I thought.
Let $A$ be a Borel subset of $\mathbb R$. We define $$ \begin{align*} B &:= \{(s, t) \in \mathbb R^2 : s \le t \le s+1 \text{ and } t \in A\} , \\ C &:= \{(s, t) \in \mathbb R^2 : s \le t \le s+1 \text{ and } s \in A\}. \end{align*} $$
Could you provide some hints to prove that $B,C$ are Borel subsets of $\mathbb R^2$?
Let $D = \{ (s,t) | s \le t \le s+1 \}$, it is straightforward to see that $D$ is Borel. Similarly we can show that $\mathbb{R} \times A$ is Borel. Then $B = D \cap (\mathbb{R} \times A) $ is Borel.