On some analogies between measure and category and their applications in the theory of additive functions
Abstract
The first part of the paper is devoted to topological analogies of some theorems from real analysis (Vitali’s covering theorem, Lebesgue theorem on outer density, Smital’s lemma). Then we give a topological analogue of a theorem of Ostrowski connected with additive functions.
In the last part we deal with Hamel bases, specially with Burstin bases. We prove their existence and study topological and measure properties.
References
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2. J. Aczél, Lectures on functional equations and their applications, Academic Press, New York, 1966.
3. S. Banach, Théorème sur les ensembles de première catégorie, Fund. Math. 16 (1930), 395-398.
4. C. Burstin, Die Spaltung des Kontinuums in c im L. Sinne nichtmessbare Mengen, Sitzungsber. Akad. Wiss. Wien, Math. nat. Klasse, Abl. IIa 125 (1916), 209-217.
5. A. Császár, Konvex halmazokról és függvényekröl, Mat. Lapok 9 (1958), 273-282.
6. R. Ger, Note on convex functions bounded on regular hypersurfaces, Demonstratio Math. 6 (1973), 97-103.
7. R. Ger, Thin sets and convex functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 413-416.
8. R. Ger, M. Kuczma, On the boundedness and continuity of convex functions and additive functions, Aequationes Math. 4 (1970), 157-162.
9. I. Halperin, Nonmeasurable sets and the equation f(x+ y) = f(x)+f(y), Proc. Amer. Math. Soc. 2 (1951), 221-224.
10. G. Hamel, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung f(x+ y) = f(x)+f(y), Math. Ann. 60 (1905), 459-462.
11. Z. Kominek, On the sum and difference of two sets in topological vector spaces, Fund. Math. 71 (1971), 165-169.
12. M. Kuczma, Additive functions and the Egorov theorem, General Inequalities 1 [Proc. 1st International Meeting on General Inequalities, Oberwolfach 1976], Edited by E.F. Beckenbach, Birkhäuser Verlag, Basel, 1978, 169-173.
13. M. Kuczma, J. Smítal, On measures connected with the Cauchy equation, Aequationes Math. 14 (1976), 421-428.
14. K. Kuratowski, Topologie I. Monografie Mat. 20, Polish Scientific Publishers, Warszawa, 1958.
15. S. Marcus, Généralisation, aux fonctions de plusieurs variables, des théorèmes de Alexander Ostrowski et de Masuo Hukuhara concernant les fonctions convexes (J), J. Math. Soc. Japan. 11 (1959), 171-176.
16. M.R. Mehdi, Some remarks on convex functions, J. London Math. Soc. 39 (1964), 321-326.
17. A. Ostrowski, Über die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichungen, Jahresber. Deutsch. Math.-Verein. 38 (1929), 54-62.
18. J.C. Oxtoby, Measure and category, Springer Verlag, New York-Heidelberg-Berlin, 1971.
19. R.A. Rosenbaum, Subadditive functions, Duke Math. J. 17 (1950), 227-242.
20. W. Sierpiński, Sur la question de la mesurabilité de la base de Hamel, Fund. Math. 1 (1920), 105-111.
KuczmaM. (1985). On some analogies between measure and category and their applications in the theory of additive functions. Annales Mathematicae Silesianae, 1, 155-162. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14346
Marek Kuczma
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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