= Bracketing of a function class
{title2=$[\ell,u],\quad\|u-\ell\|_{L^1(P)}$}
= Function bracket
{synonym}
A bracket $[\ell,u]$ contains every measurable function $h$ satisfying $\ell\le h\le u$ pointwise. For a probability measure $P$, its $L^1(P)$ width is $P(u-\ell)$, with integrable endpoints. Finite bracketing at every positive width is a sufficient complexity condition for a <uniform law of large numbers>. Pointwise inequalities, or inequalities outside a single common null set, allow empirical inequalities to hold simultaneously throughout the class.
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