Connexion en topologie fine et balayage des mesures
Annales de l'Institut Fourier, Volume 21 (1971) no. 3, p. 227-244

The fine topology is shown to be connected and locally connected in the case of a harmonic space Ω satisfying the group of axioms (A 1 ) in Brelot’s theory (thus including the domination axiom). Another main result (though established here, in its entirety, for the classical case of a Green space only) asserts that, for every positive measure μ on Ω, say of compact support, and for any base BΩ such that μ(B)=0, the fine support of the swept-out measure μ B coincides with the fine boundary of the union of all those fine components of the complement of B which are charged by μ.

On montre d’abord que la topologie fine est connexe et localement connexe, dans le cas d’un espace harmonique Ω satisfaisant au groupe d’axiomes (A 1 ) de Brelot (y compris l’axiome de domination). Un autre résultat principal (qu’on n’établit complètement ici que pour le cas classique d’un espace de Green) affirme que, pour toute mesure positive μ sur Ω, soit à support compact, et pour toute base BΩ telle que μ(B)=0, la mesure balayée μ B a pour support fin la frontière fine de la réunion de toutes les composantes fines du complémentaire de B chargées par μ.

@article{AIF_1971__21_3_227_0,
     author = {Fuglede, Bent},
     title = {Connexion en topologie fine et balayage des mesures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {21},
     number = {3},
     year = {1971},
     pages = {227-244},
     doi = {10.5802/aif.388},
     zbl = {0208.13802},
     mrnumber = {49 \#9241},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1971__21_3_227_0}
}
Fuglede, Bent. Connexion en topologie fine et balayage des mesures. Annales de l'Institut Fourier, Volume 21 (1971) no. 3, pp. 227-244. doi : 10.5802/aif.388. http://www.numdam.org/item/AIF_1971__21_3_227_0/

[1] M. Brelot, Points irréguliers et transformations continues en théorie du potentiel, J. de Math. (Liouville), (1940), 319-337. | JFM 66.0447.01 | MR 3,47b | Zbl 0024.40301

[2] M. Brelot, Quelques propriétés et applications du balayage, C.R. Acad. Sci. (Paris), 227, (1948), 19-21. | MR 10,116f | Zbl 0038.26203

[3] M. Brelot, Lectures on potential theory. Tata Institute of Fundamental Research. Bombay (1960). | MR 22 #9749 | Zbl 0098.06903

[4] M. Brelot, Axiomatique des fonctions harmoniques. Montréal (1966). | Zbl 0148.10401

[5] M. Brelot et G. Choquet, Espaces et lignes de Green. Ann. Inst. Fourier, 3, (1951), 199-263. | Numdam | MR 16,34e | Zbl 0046.32701

[6] G. Choquet, Démonstration non probabiliste d'un théorème de Getoor, Ann. Inst. Fourier 15, 2 (1965), 409-414. | Numdam | MR 33 #5929 | Zbl 0141.30501

[7] C. Constantinescu, Some properties of the balayage of measures on a harmonic space, Ann. Inst. Fourier, 17, (1967), 273-293. | Numdam | MR 37 #3033 | Zbl 0159.40804

[8] A. Denjoy, Sur les fonctions dérivées sommables, Bull. Soc. Math. France, 43, (1916), 161-248. | JFM 45.1286.01 | Numdam

[9] J.L. Doob, Applications to analysis of a topological definition of smallness of a set, Bull. Amer. Math. Soc., 72, (1966), 579-600. | MR 34 #3514 | Zbl 0142.09001

[10] B. Fuglede, Propriétés de connexion en topologie fine. Prépublication. Copenhague (1969).

[11] B. Fuglede, Fine connectivity and finely harmonic functions, C.R. Congr. Internat. Math., Nice (1970). | Zbl 0223.31016

[12] R.-M. Herve, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, 12, (1962), 415-571. | Numdam | MR 25 #3186 | Zbl 0101.08103

[13] Ng.-Xuan-Loc and T. Watanabe, A characterization of fine domains for a certain class of Markov processes with applications to Brelot harmonic spaces. (To appear).

[14] J. Ridder, Über approximativ stetigen Funktionen. Fund. Math., 13, (1929), 201-209. | JFM 55.0145.01