Originally Posted by

**Jose27** This works fine, but you would have to use one of the following results:

1) If $\displaystyle \mu (X) < \infty$ then $\displaystyle L^p (X) \subset \L^q (X)$ for all $\displaystyle 1\leq q \leq p$

2) If $\displaystyle f\in L^p (X)$ ,$\displaystyle g \in L^q (X)$ where $\displaystyle \frac{1}{p} +\frac{1}{q} =1$ then $\displaystyle fg\in L^1 (X)$ and the for fixed $\displaystyle f$, the map $\displaystyle g\mapsto \int_X fg $is continous (Riesz representation theorem).

Another approach that works for arbitrary $\displaystyle X$ and $\displaystyle p$ is taking $\displaystyle A_m:= \{ f\in L^p (X) : f\geq 0 \, \ f(x)\leq m \ \ \mbox{and} \ \int_X f <\infty \}$ and using the monotone convergence theorem on $\displaystyle f_m = f|_{A_m}$.