Uniform conditional L2 norm ($\Vert f{\Vert }_{2,\infty \mid Y}={\mathrm{ess}\sup }_y\Vert f{\Vert }_y$)

The essential supremum over the base of the conditional L2 norm. It may be infinite for a globally square-integrable function. For probability fibers, $\Vert f{\Vert }_2\le \Vert f{\Vert }_{2,\infty \mid Y}$; the uniform norm controls all fibers outside a null set rather than only their average.