For a Banach space X, L Φ (T, X) denotes the metric space of all X-valued Φ-integrable functions f : T → X , where the measure space (T, , µ) is a complete positive σ-finite and Φ is an increasing subadditive continuous function on [0, ∞) with Φ (0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y ⊆ X to a finite number of elements in X