Let $f : \mathbb{R} \to (0,\infty)$ be a Borel measurable function and let $E$ be a Borel measurable subset of $\mathbb{R}$ such that $\lambda(E)>0$ . Denote $ F(t) = \int_E f(t + x)d\lambda(x), t \in \mathbb{R}. $
Prove that $F$ is Borel measurable.
Prove that if $F \in L^1(\mathbb{R})$, then $f \in L^1(\mathbb{R})$ and $\lambda(E) < \infty.$
This question is asked similary in the below links but as of the first part I could not understand how to prove the borel measurability. I'm quite a begginer in real analysis, so I was wondering if someone can help by through a step by step understanding of the proofs suggested in the links.