Inclusion in all $L^p$ spaces

Question

If you can show that a random variable is in every $L^p$-space for a given measure, for every $1<p<\infty$ then can you deduce that it is also in $L^1$?

Answer 1

Of course. If $\mu$ is a finite postive measure and $f\in L^2(\mu),$ then

$$\int_X|f|\,d\mu = \int_X|f|\cdot 1\,d\mu \le (\int_X|f|^2\,d\mu)^{1/2}\cdot (\int_X 1^2\,d\mu)^{1/2}$$ $$ = (\int_X|f|^2\,d\mu)^{1/2}\cdot \mu(X)^{1/2} < \infty.$$