Convergence in measure and in category. (English) Zbl 1465.28002
Summary: W. Orlicz [Stud. Math. 12, 286–307 (1951; Zbl 0044.05701)] has observed that if \(\{f_n( \cdot, y)\}_{n\in N}\) converges in measure to \(f(\cdot, y)\) for each \(y \in [0, 1]\), then \(\{f_n\}_{n \in N}\) converges in measure to \(f\) on \([0, 1] \times [0, 1]\). The situation is different for the convergence in category even if we assume the convergence in category of sequences \(\{(f_n (\cdot, y)\}_{n \in N}\) for each \(y \in [0, 1]\) and \(\{f_n (x, \cdot)\}_{n\in_N}\) for each \(x \in [0, 1]\).
MSC:
| 28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |
Citations:
Zbl 0044.05701References:
| [1] | R.M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002. · Zbl 1023.60001 |
| [2] | E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Berlin, 1969. · Zbl 0225.26001 |
| [3] | W. Orlicz, On a class of asymptotically divergent sequences of functions, Studia Math. 12 (1951), 286-307; Collected papers, part I, 630-651, Polish Scientific Publishers, Warszawa, 1988. · Zbl 0044.05701 |
| [4] | J.C. Oxtoby, Measure and Category, second edition, Springer-Verlag, New York-Berlin, 1980. · Zbl 0435.28011 |
| [5] | E. Wagner, Sequences of measurable functions, Fund. Math. 112 (1981), 89-102. · Zbl 0386.28005 |
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