On Zolezzi's Theorem for Infinite Measure Spaces

We discuss the infinite measure counterpart of Zolezzi's Theorem for infinite measure spaces. For a measure space with infinite measure, $(\Omega, \Sigma, \mu)$, we construct a sequence in $L^\infty(\mu)$, with uniformly control upon its support measure, that does not converge in $L^p(\mu)$, for all $1\le p < \infty$, however does converge weakly in $L^\infty(\mu)$.