# Optimal Domains and Integral Extensions

Chapter

## 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. L^{1}(m_{T}) which is σ-o.c. and has the desirable property that every function from X(μ) belongs to L^{1}(M_{T}) and$$
T(f) = \int_\Omega {f dm_T } , f \in X(\mu ) \subseteq L^1 (m_T ).
$$

## Keywords

Banach Space Banach Lattice Vector Measure Fredholm Operator Continuous Linear Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Birkhäuser Verlag AG 2008