An elementary proof of Komlós-Révész theorem in Hilbert spaces. (English) Zbl 0898.46027
The paper contains an elementary proof of the following theorem:
If \((f_n)\) is a bounded sequence in \(L^1_H\), then there exists a subsequence \((g_n)\) of the sequence \((f_n)\) and a \(\mu\)-integrable function \(g\) such that \[ {1\over k} \sum^k_{n= 1} h_n(w)\to g(w),\quad \mu\text{-almost everywhere, for }k\to\infty, \] for each subsequence \((h_n)\) of the sequence \((g_n)\).
Let us remark, that in the course of the proof, the author uses only elementary mathematical tools.
If \((f_n)\) is a bounded sequence in \(L^1_H\), then there exists a subsequence \((g_n)\) of the sequence \((f_n)\) and a \(\mu\)-integrable function \(g\) such that \[ {1\over k} \sum^k_{n= 1} h_n(w)\to g(w),\quad \mu\text{-almost everywhere, for }k\to\infty, \] for each subsequence \((h_n)\) of the sequence \((g_n)\).
Let us remark, that in the course of the proof, the author uses only elementary mathematical tools.
Reviewer: A.Waszak (Poznań)
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |
