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,

, we construct a sequence in

L^\infty(\mu) L ∞ ( μ )

, with uniformly control upon its support measure, that does not converge in

L^p(\mu) L p ( μ )

, for all

1\le p < \infty 1 ≤ p < ∞

, however does converge weakly in

L^\infty(\mu) L ∞ ( μ )

.