L ^p -valued Measures Without Finite X-semivariation for 2 < p < ∞

Abstract

We show that for 1 ≤ p < ∞, the property that every L ^p -valued vector measure has finite X-semivariation in L ^p (μ, X) is equivalent to the property that every continuous linear map from l ¹ to X is p-summing. For 2 < p < ∞, we explicitly construct an L ^p([0, 1])-valued measure without finite L ^p -semivariation.