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Given a Schauder basis in a Banach function space of a specified type, one can delete any finite number of elements of the basis and still preserve serial totality by making an arbitrarily small adjustment of the domain. Let {0b},?= be a system of real valued functions finite almost everywhere and measurable on a set Gc [0, 1], mes. G>0. Talalyan [2], [3] proved the following to be equivalent: (a) {On1}n=1 iS total in measure on G, that is, for every measurable function f defined on G, there exists a sequence of finite linear combinations of functions of the system {bn }n=L which converges in measure tof on G. (b) For each positive number 8, there is a measurable set Se whose measure exceeds 1 -8, such that {On }n=l is total in L2(S,). We shall say that {0bk}'=l iS serially total in some function space, if for any given functionf in the space we can find a series ak?bk which converges tof in the metric of the space. The result of this paper can be viewed as a first step in changing total into serially total. Let L(E) be a Banach space of measurable functions on a measurable set Ec [a, b] with natural linear operations. As usual, identify functions equal almost everywhere. Postulate the following on L(E): (1) L(E) is contained in L1(E); (2) L(E) contains the function 1; (3) iff E L(E), and iffor a measurable function g, O?g(x)_?f(x) almost everywhere, then g E L(E); (4) if f E L(E) and x_A is the characteristic function of the measurable set A, then UIIfx411 Ilf IlI goes to zero as JAI goes to zero, where JAI denotes the Lebesgue measure of A. Example of spaces that satisfy (1)-(4) are the LP spaces ?

On the serial completion of deleted Schauder bases by domain adjustment