Expectation of stopped uniform AMARTS and convergence of MILS and GFTs (Q909340)

The paper under review consists of two parts. In the first part, the author shows that for every uniform amart \(\{X_ n\}\) in a Banach space satisfying \(\sup_ N\int_{\Omega}\| X_ n\| =\infty\) there exists a stopping time \(\tau\) such that \(\int_{\{\tau <\infty \}}\| X_{\tau}\| =\infty.\) This result is known and even valid for semi- amarts; see Corollaire 2.4 and Théorème 3.2 of the reviewer's paper, C. R. Acad. Sci., Paris, Sér. A 288, 431-434 (1979; Zbl 0397.60008). The author also claims authorship of the corresponding result for real- valued semi-amarts, which is the Corollaire resp. Corollary 3.3 of the reviewer's papers ibid. 287, 663-665 (1978; Zbl 0394.60045), resp. J. Multivariate Anal. 10, 123-134 (1980; Zbl 0418.60045). None of these papers is quoted in the references. In the second part, the author studies almost sure convergence and mainly proves a minor variant of a result of \textit{M. Talagrand} [Ann. Probab. 13, 1192-1203 (1985; Zbl 0582.60055)] on the convergence of a mil in a Banach space having the Radon-Nikodym property.