I am starting to take an interest in convex geometry and stumbled on the following theorem due to Pisier, the proof should be in https://link.springer.com/content/pdf/10.1007/BF01450929.pdf, although the theorem below is rephrased.
Let $X$ be a subspace of $L_1(\Omega,\mu)$ of type $p$ for some $p>1$. Then, there is a non-negative function $F$ in $L_1(\Omega,\mu)$ such that
$\|F\|_{L_1(\Omega, \mu)} = 1, \{w; F(w) = 0\} \subseteq \cap_{x \in X}\{w; x(w) = 0\}$
and so that for every $x \in X$
$\|x(w)/F(w)\|_{L_{p, \infty}(\Omega, F\mu)} \leq e T_p(X)\|x\|_{L_1(\Omega, \mu)}$.
Here, $\|f\|_{p, \infty} = \sup t(\mu\{\omega \in \Omega: |f(\omega)|>t\})^{1/p} < \infty$. (on some probability measure space $(\Omega, \mu)$ ).
I already have difficulty parsing this inequality. Specifically, it is not clear to me what is meant by $\|x(w)/F(w)\|_{L_{p, \infty}(\Omega, F\mu)}$. $F\mu$ is not necessarily a probability distribution. The only thing I could imagine is something of the sort of
$\|x(w)/F(w)\|_{L_{p, \infty}(\Omega, F\mu)} = \sup t [\int \mathbb{1}_{\{\omega \mid |x(w)/F(w) > t\}} F(w) d\mu(\omega)]^{1/p}$.
What is meant by $\|x(w)/F(w)\|_{L_{p, \infty}(\Omega, F\mu)}$? Can you give me an example of such a space and such a function $F$? Is there some sketch of a proof (that uses no heavy notation from functional analysis) for me to make sense of it?