Vector Measures and Integrals

Abstract

From here up to section 1.3 (Ω, Σ) is an abstract measurable space, X, Y, Z are the Banach spaces. For µ ∈ M(Ω, X)we put

$$\left| \mu \right|:\sum { \to \left[ {0,\infty } \right]} :\left| \mu \right|(A) = \sup \sum\limits_i {\left\| {\mu ({A_i})} \right\|} $$

where sup is taken over all finite partitions of the set $A:A = \mathop \cup \limits_{i = 1}^n {A_i}$ , ${A_i} \cap {A_j} = \emptyset$ A _i ∈ Σ The measure µ ∈ M(Ω, X) is called bounded if ∣µ∣(Ω) < ∞ we denote by MB(Ω X) the totality of such measures.