# Primitives and Integrals

Chapter

## Abstract

Unless expressly mentioned to the contrary, in this chapter we shall only consider vector functions of a*real* variable which take their values in a *complete* normed space over **R**. When we deal in particular with real-valued functions it will always be understood that these functions are *finite* unless stated to the contrary.

## Keywords

Compact Subset Vector Function Regulate Function Step Function Real Function
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## References

- 1.Recall (
*Set Theory*, III, p. 144) that a set a of subsets of I is*directed with respect to the relation*⊂ if, for any X ∈ F, Y ∈ F, there exists Z ∈ F such that X ⊂ Z and Y ⊂ Z. If S(X) denotes the subset of F formed by the U ∈ F such that*U*⊃*X*, then the S(X) form a base for a filter on F, called the*filter of sections*of F the limit (if it exists) of a map*f*of F into a topological space, with respect to the filter of sections of F, is called the*limit off with respect to the directed set*F (*cf. Gen. Top*., I, p. 70 and*Gen. Top*., IV, p. 348).Google Scholar

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© Springer-Verlag Berlin Heidelberg 2004