Let (XxY, SxT, μxν) denote the completion of the Cartesian product of the σ-finite and complete "measure spaces (X, S, μ.) and (Y, T, ν) [3]. Let λx and λy denote arbitrary length functions defined on (X, S, μ.) and (Y, T, ν) respectively, the conjugate length functions [2]. We suppose that

1

is defined for every f(x, y) measurable (SxT). The Fubini theorem implies that f(x, y) is measurable (T) for almost all x. Thus λ xy(f) will be defined when λy(f) Is measurable (S). If Lλ y = Lp, 1 ≤ p < ∞, this is implied by the Fubini theorem. General conditions ensuring that λ y (f) is measurable (S) are given in [l, Theorem 3.2] When λxy(f) is defined for every f(x, y) measurable (SxT), it is a length function and Lλ xy is a Banach space [l, Theorem 3.l].