Let E be a compact of C N and let L p be a modular functions space. In this paper we consider the problem of analytic extension of f ϵ L p to a holomorphic function defined on an open neighborhood of E. In particular, we generalize some of the results obtained in [W. Ples̀niak, “Quasianalyticity in F-spaces of Integrable Functions, Approximation, and Function Spaces” (Z. Ciesielski, Ed.), pp. 553–571, Proceedings of the International Conference held in Gdánsk, August 27–31, PWN Warszawa, North-Holland, Amsterdam 1979; W. Ples̀niak, “Leja's Type Polynomial Condition and Polynomial Approximation in Orlicz Spaces,” Ann. Polon. Math. 46 (1985), 268–278] for the case of Orlicz Spaces and in [W. M. Kozlowski and G. Lewicki, “On Polynomial Approximation in Modular Function Spaces” (J. Musielak, Ed.), pp. 63–68, Proceedings of the International Conference “Function Spaces” held in Poznań, August 1986, Teubner, Stuttgart, 1988] for the case of s-convex function modulars.