A Banach principle for \(L^ \infty\) (Q1921271)

In the context of the spaces \(L^p\) with \(1\leq p<\infty\), the classical Banach principle relates the a.e. convergence of a sequence of operators and the finiteness of the corresponding maximal function. The authors formulate a version of this principle for \(L^\infty\); it relates the a.e. convergence of a sequence of operators and the continuity of the maximal function at zero on the ball of \(L^\infty\) if this ball is endowed with the topology of convergence in measure. Some applications are given.