Let $f:(0;\infty)\times (0;\infty) \rightarrow \Bbb R$ be a Lebesgue measurable function, is $T_f(y)=\int_0^{+\infty} f(x,y) \, dy$ measurable?

Question

Let $f:(0;\infty)\times (0;\infty) \rightarrow \Bbb R$ be a Lebesgue measurable function. Lets define $$T_f(y)=\int_0^{+\infty} f(x,y) \, dy$$ $$G_f(y)=\int_0^{+\infty} |f(x,y)| \, dy$$ By Tonelli's Theorem I know $G_f$ is measurable and I also know $G_f \in L^p(0;+\infty)$ (take it as an hypothesis). I would like to say $\Vert T_f \Vert _p \le \Vert G_f \Vert _p$ so $T_f \in L^p(0;+\infty)$ but I need $T_f$ to be measurable in order to be abel to calculate $\Vert T_f \Vert _p$. Is this true? How can I prove it?