Pseudo-Differential Operators and Symmetries pp 115-189 | Cite as

# Measure Theory and Integration

Chapter

## Abstract

This chapter provides sufficient general information about measures and integration for the purposes of this book. The starting point is the concept of an outer measure, which “measures weights of subsets of a space”. We should first consider how to sum such weights, which are either infinite or non-negative real numbers. For a finite set abbreviates the usual sum of numbers Infinite summations are defined by limits as follows: Show that

*K*, notation$$
\sum\limits_{j \in K} {a_j }
$$

*a*_{ j }∈ [0, ∞] over the index set*K*. The conventions here are that*a*+∞=∞ for all*a*∈ [0, ∞], and that$$
\sum\limits_{j \in 0/} {a_j = 0.}
$$

**Definition C.0.1.**The*sum*of numbers*a*_{ j }∈ [0, ∞] over the index set*J*is$$
\sum\limits_{j \in J} {a_j : = } \sup \left\{ {\sum\limits_{j \in K} {a_j :} K \subset J is finite} \right\}.
$$

**Exercise C.0.2.**Let 0 <*a*_{ j }< ∞ for each*j*∈*J*. Suppose$$
\sum\limits_{j \in J} {a_j } < \infty .
$$

*J*is at most countable.## Keywords

Measure Space Signed Measure Measure Theory Outer Measure Disjoint Family
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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© Birkhäuser Verlag AG 2010