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**A note on convergence in measure.**
*(English)*
Zbl 0941.28002

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 \(h(f_n,g_n)\to h(f,g)\)?

Reviewer: I.Netuka (Praha)

### MSC:

28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |

26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |