Ways of obtaining topological measures on locally compact spaces
Topological measures and quasi-linear functionals generalize measures and linear functionals. Deficient topological measures, in turn, generalize topological mea- sures. In this paper we continue the study of topological measures on locally compact spaces. For a compact space the existing ways of obtaining topological measures are (a) a method using super-measures, (b) composition of a q-function with a topological measure, and (c) a method using deficient topological measures and single points. These techniques are applicable when a compact space is connected, locally connected, and has a certain topological characteristic, called “genus”, equal to 0 (intuitively, such spaces have no holes). We generalize known techniques to the situation where the space is locally compact, connected, and locally connected, and whose Alexandroff one-point compactification has genus 0. We define super-measures and q-functions on locally com- pact spaces. We then obtain methods for generating new topological measures by using super-measures and also by composing q-functions with deficient topological measures. We also generalize an existing method and provide a new method that utilizes a point and a deficient topological measure on a locally compact space. The methods presented allow one to obtain a large variety of finite and infinite topological measures on spaces such as Rn, half-spaces in Rn, open balls in Rn, and punctured closed balls in Rn with the relative topology (where n ≥ 2).
1. Aarnes J.F. Quasi-states and quasi-measures. Adv. Math., 1991, vol. 86, no. 1, pp. 41–67. https://doi.org/10.1016/0001-8708(91)90035-6
2. Aarnes J.F. Pure quasi-states and extremal quasi-measures. Math. Ann., 1993, vol.295, pp. 575–588. https://doi.org/10.1007/BF01444904
3. Aarnes J.F. Construction of non-subadditive measures and discretization of Borel measures. Fundamenta Mathematicae, 1995, vol. 147, pp. 213–237. https://doi.org/10.4064/fm-147-3-213-237
4. Aarnes J.F., Butler S.V. Super-measures and finitely defined topological measures. Acta Math. Hungar., 2003, vol. 99 (1-2), pp. 33–42. https://doi.org/10.1023/A:1024549126938
5. Aarnes J.F., Rustad A.B. Probability and quasi-measures – a new interpretation. Math. Scand., 1999, vol. 85, no. 2, pp. 278–284. https://doi.org/10.7146/math.scand.a-18277
6. Butler S.V. Q-functions and extreme topological measures, J. Math. Anal. Appl., 2005, vol. 307, pp. 465–479. https://doi.org/10.1016/j.jmaa.2005.01.013
7. Butler S.V. Solid-set functions and topological measures on locally compact spaces, submitted.
8. Butler S.V. Deficient topological measures on locally compact spaces, submitted.
9. Grubb D.J. Irreducible Partitions and the Construction of Quasi-measures. Trans. Amer. Math. Soc., 2001, vol. 353, no. 5, pp. 2059–2072. https://doi.org/10.1090/S0002-9947-01-02764-7
10. Johansen Ø., Rustad A. Construction and Properties of quasi-linear func- tionals. Trans. Amer. Math. Soc., 2006, vol. 358, no. 6, pp. 2735–2758. https://doi.org/10.1090/S0002-9947-06-03843-8
11. Knudsen F.F. Topology and the construction of Extreme Quasi-measures. Adv. Math., 1996, vol. 120, no. 2, pp. 302–321. https://doi.org/10.1006/aima.1996.0041
12. Entov M., Polterovich L. Quasi-states and symplectic intersections. ArXiv, 2004.
13. Polterovich L., Rosen D. Function theory on symplectic manifolds. CRM Monograph series, vol. 34, American Mathematical Society, Providence, Rhode Island, 2014. https://doi.org/10.1090/crmm/034
14. Svistula M.G. A Signed quasi-measure decomposition. Vestnik SamGU. Estestvennonauchnaia seriia, 2008, vol. 62, no. 3, pp. 192–207. (In Russian).
15. Svistula M.G. Deficient topological measures and functionals generated by them. Sbornik: Mathematics, 2013, vol. 204, no. 5, pp. 726–761. https://doi.org/10.1070/SM2013v204n05ABEH004318