My question is analogous to the embedding $L^p\subset L^q(\Omega)$, for $p>q$ and for a bounded $\Omega$. In weak $L^p$ spaces, that is, $L^{p,\infty}$, does such an inclusion hold for arbitrary sets, where
$|||f|||_{L^{p,\infty}}^p=\sup_{\lambda>0}\left\{\lambda^p\mathcal{L}^N\left\{|f|>\lambda\right\}\right\}$
I would hope that an estimate like this would hold: $|||f|||_{L^{p,\infty}}\leq |||f|||_{L^{q,\infty}} $ for $p<q$. I have no counterexample. This is a little tricky because it defines a pseudonorm, not a norm.
Thank you to whoever answers.
If $\lambda\le1$ then $|\{|f|>\lambda\}|\le|\Omega|$. If $\lambda>1$ then $$ |\{|f|>\lambda\}|\le\frac{\|f\|_{p,\infty}}{\lambda^p}\le\frac{\|f\|_{p,\infty}}{\lambda^q}. $$