If xf(x) is Lebesgue integrable in R+, how do I show that:
f(x)exp(-tx) is integrable for every t and x in R+?
For the exercise 9.1 I've been trying to use the conditions of mesurability of f(x)exp(-tx) which is okay. Showing that f(x)exp(-tx) is continue in relation to t and what it's needed is just to show integrability of f(x)exp(-tx) to be sure that F(x) is defined.
In your exercise it's given that $f\in\mathcal{L}^1(\Bbb R_+)$ hence we have $$\int_{\Bbb R_+} |f(x)| dx < \infty$$
Additionally for each $$t,x \in \Bbb R_+$$ we have $$e^{-tx} \le 1$$ and we get:
$$\int_{\Bbb R_+} |f(x)e^{-tx}| dx\quad \le \quad \int_{\Bbb R_+} |f(x)| dx \quad < \quad \infty$$ what gives the wanted result.