Dense Sets in Spaces of Integrable Functions

Abstract

In this paper, applying a modified version of the Stone-Weierstrass theorem, an approximation theorem is proved for spaces of measurable complex functions whose powers of their absolute values are integrable on a Hausdorff space, being the countable union of compact subsets, with respect to a finite and regular Borel measure. In this theorem a family of bounded measurable functions, among which at most countably many may be discontinuous, is used for approximation. It is proved that if such family contains together with a function also its conjugation, and if the family separates almost everywhere the points of the integration domain, then the linear space spanned by the set of products of powers of the functions belonging to the family is dense in spaces of integrable functions. This theorem is also generalized to families which may contain uncountably many discontinuous functions and on σ-finite measures.