= Weak Lq space
{title2=$L^{q,\infty}$}
For $1<q<\infty$, the weak Lq space consists of measurable functions for which $A_q(f)=\sup_{t>0}t|\{|f|>t\}|^{1/q}$ is finite. This distribution-function expression is a <quasi-norm>. The equivalent norm $N_q(f)=\sup_{0<|E|<\infty}|E|^{-1/q'}\int_E|f|$ satisfies $A_q\leq N_q\leq q'A_q$, with $q'=q/(q-1)$. The <layer cake representation> bounds the integral by $\int_0^\infty\min(|E|,(A_q/t)^q)dt=q'A_q|E|^{1/q'}$. The lower bound uses finite-measure subsets of a superlevel set.
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