Optimal Domains and Integral Extensions

Abstract

In the previous chapter we presented a selection of relevant aspects from the theory of vector measures and integration. Following a time honoured practice, we began with a vector measure ? and ended up with operators defined on the spaces L^p(v) and L^p/_w(v). In this chapter we reverse this line of development in a certain sense. Namely, we begin with an operator T : X(_μ) → E, defined on some σ-order continuous q-B.f.s. X(_μ) and taking values in a Banach space E, and produce from it the E-valued vector measure m_T : A → T(X_A). This, in turn, has associated with it the B.f.s. L¹(m_T) which is σ-o.c. and has the desirable property that every function from X(μ) belongs to L¹(M_T) and

$$ T(f) = \int_\Omega {f dm_T } , f \in X(\mu ) \subseteq L^1 (m_T ). $$