A continuous version of Liapunov's convexity theorem
Annales de l'I.H.P. Analyse non linéaire, Tome 5 (1988) no. 1, pp. 23-36. 
@article{AIHPC_1988__5_1_23_0,
     author = {Cellina, Arrigo and Colombo, Giovanni and Fonda, Alessandro},
     title = {A continuous version of {Liapunov's} convexity theorem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {23--36},
     publisher = {Gauthier-Villars},
     volume = {5},
     number = {1},
     year = {1988},
     mrnumber = {936888},
     zbl = {0647.28002},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPC_1988__5_1_23_0/}
}
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Cellina, Arrigo; Colombo, Giovanni; Fonda, Alessandro. A continuous version of Liapunov's convexity theorem. Annales de l'I.H.P. Analyse non linéaire, Tome 5 (1988) no. 1, pp. 23-36. http://archive.numdam.org/item/AIHPC_1988__5_1_23_0/

[1] A. Fryszkowski, Continuous Selections for a Class of Non-Convex Multivalued Maps, Studia Mathematica, T. LXXVI, 1983, pp. 163-174. | MR | Zbl

[2] P. Halmos, The Range of a Vector Measure, Bull. Am. Math. Soc., Vol. 54, 1948, pp. 416-421. | MR | Zbl

[3] P. Halmos, Measure Theory, Van Nostrand, Princeton, 1950. | MR | Zbl

[4] A. Liapunov, Sur les fonctions-vecteurs complètement additives, Bull. Acad. Sci. U.R.S.S., Ser. Math., Vol. 4, 1940, pp. 465-478 (Russian). | JFM | MR | Zbl

[5] A. Liapunov, same as above, Vol. 10, 1946, pp. 277-279 (Russian); for an English version, see J. Diestel and J.J. Uhl, Vector Measures, 1977, A.M.S., Providence,Rhode Island, p. 262.

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[7] M. Valadier, Une mesure vectorielle sans atoms dont l'ensemble des valeurs est non convexe, Boll. U.M.I., Serie VI, Vol. II-C, No. 1, 1983, pp. 293-296. | MR | Zbl