On a theorem of Goffman concerning Schauder series

Abstract

1. A classical question posed by Lusin asks whether it is possible to find for each measurable function defined on [0, 2ir] a corresponding trigonometric series, with coefficients converging to zero, that converges almost everywhere to the function. This question was answered affirmatively by Men'sov [4] for real-valued functions, but the general question remains unanswered. Thus, it is of interest to inquire whether there be any Schauder bases for LP, p> l, with respect to which every measurable function has a pointwise representation. Although Talalyan and Arutyanyan [8] have shown that a prime candidate, the Haar system, does not have this property, Gundy [3 ] has proved that systems of the specified type do exist. The Schauder functions are total in each of the LP spaces, although they in no case constitute a basis. In [1], Goffman solved Lusin's problem for this system by way of a sequence of careful estimates culminating in a construction of the required series. An interesting aspect of this work is that not all of the Schauder functions are required for the construction. Indeed, it is clear from a superficial examination of the arguments employed in [1] that any finite number of functions could be deleted from the system and the work carried through with no resulting essential modification of the demonstrations. In analogy with work of Talalyan [6], and Goffman and Waterman [2], it is appropriate to ask whether infinitely many Schauder functions could be discarded in such a way that the above-mentioned result of Goffman would remain in force for the abbreviated system. In the present note it is shown that this is the case, and a characterization is given of those subsystems for which the Goffman theorem holds. The result suggests that there may be lurking in the background some very general form of the Muintz theorem.