Let $\mu$ be a finite measure on a measurable space $(X,\Sigma)$. I want to prove that exists $C > 0$ so that $$ \| f\|_p \leq C \| f \|_{q,\infty}$$ when $1 \leq p < q < \infty$, where $$ \| f \|_{q,\infty} = \sup\limits_{\lambda > 0}\lambda \mu(\{x \in X: |f(x)| > \lambda \} )^{1/q}. $$ I think the key of the proof relies in the fact that $\mu$ is a finite measure, because it's not difficult to show a counterexample when you consider the Lebesgue measure on $\mathbb{R}$.
My best aproach consists of putting $$ \int_X |f|^p d\mu = \int_0^\infty \mu\{ |f| > t\} p t^{p-1} dt = \int_0^\lambda\mu\{ |f| > t\} p t^{p-1} dt + \int_\lambda^\infty \mu\{ |f| > t\} p t^{p-1} dt, $$ where $\lambda > 0$. For the first integral it's easy to see that it's $\leq \mu(X) \frac{\lambda^p}{p} = C_1 \lambda^p$, but I'm stuck with the second integral and I don't know how to continue.
Any idea of how to continue the proof? Do yo know any different approach to the solution? Thank you very much.
For the second integral, bound $\mu\{ |f| > t\}$ by $t^{-q}\left\lVert f\right\rVert_{q,\infty}^q$ and the integral can be therefore controlled by a constant times $\lambda^{-q+p}$. Then choose $\lambda$ appropriately.