Darboux property of a non-additive set function
Sbornik. Mathematics, Tome 192 (2001) no. 7, pp. 969-978 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

As is known, the range of a finite positive non-atomic measure on a $\sigma$-algebra is a closed interval. In the present paper it is proved that this property holds also in a broad class of non-additive set functions on an $F$-algebra if the non-atomicity is replaced by the Saks property.
@article{SM_2001_192_7_a2,
     author = {V. M. Klimkin and M. G. Svistula},
     title = {Darboux property of a~non-additive set function},
     journal = {Sbornik. Mathematics},
     pages = {969--978},
     year = {2001},
     volume = {192},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_7_a2/}
}
TY  - JOUR
AU  - V. M. Klimkin
AU  - M. G. Svistula
TI  - Darboux property of a non-additive set function
JO  - Sbornik. Mathematics
PY  - 2001
SP  - 969
EP  - 978
VL  - 192
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2001_192_7_a2/
LA  - en
ID  - SM_2001_192_7_a2
ER  - 
%0 Journal Article
%A V. M. Klimkin
%A M. G. Svistula
%T Darboux property of a non-additive set function
%J Sbornik. Mathematics
%D 2001
%P 969-978
%V 192
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2001_192_7_a2/
%G en
%F SM_2001_192_7_a2
V. M. Klimkin; M. G. Svistula. Darboux property of a non-additive set function. Sbornik. Mathematics, Tome 192 (2001) no. 7, pp. 969-978. http://geodesic.mathdoc.fr/item/SM_2001_192_7_a2/

[1] Khalmosh P., Teoriya mery, IL, M., 1953

[2] Dinculeanu N., Vector measures, VEB, Berlin, 1966 | MR | Zbl

[3] Lyapunov A. A., “O vpolne additivnykh vektor-funktsiyakh”, Izv. AN SSSR. Ser. matem., 4:6 (1940), 465–478 | MR | Zbl

[4] Halmos P. R., “The range of a vector measure”, Bull. Amer. Math. Soc., 1948, no. 54, 416–421 | DOI | MR | Zbl

[5] Lindenstrauss J., “A short proof of Liapounoff's convexity theorem”, J. Math. Mech., 1966, no. 15, 971–972 | MR | Zbl

[6] Scozzafava R., “Completa additivita su opportune successioni di insiemi di una misura di probabilita semplicemente additiva e fortemente non atomica”, Boll. Un. Mat. Ital. B(7), 16:5 (1979), 639–648 | MR | Zbl

[7] Braskara Rao M., Braskara Rao K. P. S., “Charges on Boolean algebras and almost discrete spaces”, Mathematika, 1973, no. 20, 214–223 | MR | Zbl

[8] Maharam D., “Finitely additive measures on the integers”, Sankhyā Ser. A, 1976, no. 38, 44–49 | MR

[9] Olejček V., “Darboux property of finitely additive measure on $\delta$-ring”, Math. Slovaca, 27:2 (1977), 195–201 | MR | Zbl

[10] Armstrong T. E., Prikry K., “Liapounoff's theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures”, Trans. Amer. Math. Soc., 266:2 (1981), 499–514 | DOI | MR | Zbl

[11] Dobrakov I., “On submeasures, I”, Dissertationes Math., 112 (1974) | MR | Zbl

[12] Musial K., “Absolute continuity and the range of group valued measure”, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys., 21:2 (1973), 105–113 | MR | Zbl

[13] Landers D., “Connectedness properties of the range of vector and semimeasures”, Manuscripta Math., 1973, no. 9, 105–112 | DOI | MR | Zbl

[14] Constantinescu C., “The range of atomless group valued measures”, Comment. Math. Helv., 1976, no. 51, 207–213 | DOI | MR | Zbl

[15] Martellotti A., “Topological properties of the range of a group-valued finitely additive measure”, J. Math. Anal. Appl., 1985, no. 110, 411–424 | DOI | MR | Zbl

[16] Auman R., Shepli L., Znacheniya dlya neatomicheskikh igr, Mir, M., 1977 | MR

[17] Seever G. L., “Measures on $F$-spaces”, Trans. Amer. Math. Soc., 1968, no. 133, 267–280 | DOI | MR | Zbl

[18] Sikorskii R., Bulevy algebry, Mir, M., 1969 | MR

[19] Saks S., “Addition to the note on some functionals”, Trans. Amer. Math. Soc., 1933, no. 35, 967–974

[20] Yosida K., Hewitt E., “Finitely additive measures”, Trans. Amer. Math. Soc., 1952, no. 72, 46–66 | DOI | MR | Zbl