On a Lusin theorem for capacities

Wiesel, Johannes

doi:10.1090/proc/15713

On a Lusin theorem for capacities
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by Johannes Wiesel PDF

Proc. Amer. Math. Soc. 150 (2022), 1351-1360 Request permission

Abstract:

Let $X$ be a compact metric space and let $v$ be a sub-additive capacity defined on $X$. We show that Lusin’s theorem with respect to $v$ holds if and only if $v$ is continuous from above.

References

Adrianna Castaldo and Massimo Marinacci, A Lusin theorem for a class of Choquet capacities, Statist. Papers 43 (2002), no. 1, 137–142. Choquet integral and applications. MR 1883829, DOI 10.1007/s00362-001-0091-6
Donald L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 578344
Cong Xin Wu and Ming Hu Ha, On the regularity of the fuzzy measure on metric fuzzy measure spaces, Fuzzy Sets and Systems 66 (1994), no. 3, 373–379. MR 1300294, DOI 10.1016/0165-0114(94)90105-8
Peter J. Huber and Volker Strassen, Minimax tests and the Neyman-Pearson lemma for capacities, Ann. Statist. 1 (1973), 251–263. MR 356306
Jun Li and Masami Yasuda, Lusin’s theorem on fuzzy measure spaces, Fuzzy Sets and Systems 146 (2004), no. 1, 121–133. MR 2074206, DOI 10.1016/S0165-0114(03)00207-0
Nikolai Lusin, Sur les propriétés des fonctions mesurables, C. R. Math. Acad. Sci. Paris 154 (1912), no. 25, 1688–1690.
Zhaojun Zong, Feng Hu, and Xiaoxin Tian, Egoroff’s theorem and Lusin’s theorem for capacities in the framework of $g$-expectation, Math. Probl. Eng. , posted on (2020), Art. ID 1450486, 7. MR 4080316, DOI 10.1155/2020/1450486