A~note on convergence in measure
For a sequence \(\{f_n\}\) of real valued functions defined on a measure space \(f_n\to f\) means here that \(f_n\) converge in measure to a~function \(f\). It is known that, in general, \(f_n\to f\) and \(g_n\to g\) does not imply \(f_ng_n\to fg\). As shown in this note, the conclusion holds under the additional condition that \(f\) and \(g\) are almost bounded. (A~function \(f\) is said to be almost bounded if the measure of \(\{f>M\}\) is finite for a~suitable \(M\).) This turns out to be the right condition: If \(f\) is not almost bounded, there exists a~sequence \(\{f_n\}\) which converges to \(f\) in measure (even uniformly) and \(f_n^2\) does not converge to \(f^2\) in measure. A~natural metric on the space of almost bounded functions is introduced and, for a~function \(h{:} {\mathbb R}^2\to {\mathbb R}\), the following problem is studied: Under what circumstances \(f_n\to f\) and \(g_n\to g\) implies \(h(f_n,g_n)\to h(f,g)\)?
scientific article; zbMATH DE number 1450302