User:Caliburn/s/mt/Definition:Weak Lp Space/Definition 2

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space and let $p \in \openint 0 \infty$. Let $\mathcal S$ be the set of real-valued $\Sigma$-measurable functions $f \in \map {\mathcal M} {X, \Sigma, \R}$ such that:


 * $\ds \set {\alpha \paren {\map {F_f} \alpha}^{1/p} : \alpha \in \hointr 0 \infty}$ is bounded above

where $F_f$ is the survival function of $f$.

Let $\sim_\mu$ be the $\mu$-almost everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$.

We define the weak $L^p$ space $\map {L^{p, \infty} } {X, \Sigma, \mu}$ by the quotient set:


 * $\map {L^{p, \infty} } {X, \Sigma, \mu} = \mathcal S/\sim_\mu$