Question in Proof of theorem 11.22 Apostol mathematical analysis

Question

While self studying mathematical analysis from Tom apostol I have 2 question I proof of above mentioned theorem.

Question 1: How does author deduces $g_{x} $ is measurable on $\mathbb{R}$ ?

Question 2 : why $g_{x} $ belongs to $L^{2}( \mathbb{R}) $ ?

Any help will be really appreciated.

Answer 1

$g_x=f\circ h$ where $h(t)=x-t$. Sinec $h$ is continuous it is Borel measurable. Composition of two measurable functions is measurable. Hence $g_x$ is measurable.

$\int |g_x(t)|^{2}dt =\int |f(x-t)|^{2}dt =\int |f(y)|^{2} dy <\infty$ by the substitution $y=x-t$.