Let $(X, \mathscr{A})$ be a measurable space, and let $\mu$ be a measure on $(X, \mathscr{A})$ such tat $\mu(X) = 1$ suppose that $1\leq p_1 < p_2 < +\infty$.
Show that if $f$ belongs to $\mathscr{L} ^{p_2}(X,\mathscr{A}, \mu) $ then $f$ belongs to $\mathscr{L} ^{p_1}(X,\mathscr{A}, \mu) $ and satisfies $\| f\| _{p_1} \leq \| f\| _{p_2}$
Is there a typo in my exercise? For $\ell_p$ spaces it seems to be the other way around: http://en.wikipedia.org/wiki/Lp_space
Thankful for help
The assumption $\mu(X)=1$ is crucial. It's not satisfied by the set of natural numbers, hence it's not surprising it doesn't work in this case.
One case use Jensen's inequality with the convex function $x\mapsto |x|^{p_2/p_1}$.