Book contents
- Frontmatter
- Contents
- Preface
- 1 Theory of sets
- 2 Point set topology
- 3 Set functions
- 4 Construction and properties of measures
- 5 Definitions and properties of the integral
- 6 Related spaces and measures
- 7 The space of measurable functions
- 8 Linear functional
- 9 Structure of measures in special spaces
- 10 What is probability?
- 11 Random variables
- 12 Characteristic functions
- 13 Independence
- 14 Finite collections of random variables
- 15 Stochastic processes
- Index of notation
- General index
5 - Definitions and properties of the integral
Published online by Cambridge University Press: 05 November 2011
- Frontmatter
- Contents
- Preface
- 1 Theory of sets
- 2 Point set topology
- 3 Set functions
- 4 Construction and properties of measures
- 5 Definitions and properties of the integral
- 6 Related spaces and measures
- 7 The space of measurable functions
- 8 Linear functional
- 9 Structure of measures in special spaces
- 10 What is probability?
- 11 Random variables
- 12 Characteristic functions
- 13 Independence
- 14 Finite collections of random variables
- 15 Stochastic processes
- Index of notation
- General index
Summary
What is an integral?
Historically the concept of integration was first considered for real functions of a real variable where either the notion of ‘the process inverse to differentiation’ or the notion of ‘area under a curve’ was the starting point. In the first case a real number was obtained as the difference of two values of the ‘indefinite’ integral, while the second case corresponds immediately to the ‘definite’ integral. The so-called ‘fundamental theorem of the integral calculus’ provided the link between the two ideas. Our discussion of the operation of integration will start from the notion of a definite integral, though in the first instance the ‘interval’ over which the function is integrated will be the whole space. Thus, for ‘suitable’ functions f: Ω → R* we want to define the integral I(f) as a real number. The ‘suitable’ functions will be called integrable and I(f) will be called the integral of f.
Before defining such an operator I, we examine the sort of properties I should have before we would be justified in calling it an ‘integral’. Suppose then that A is a class of functions f: Ω → R*, and I:A → R defines a real number for every f∈A. Then we want I to satisfy:
(i) f∈A, f(x) ≥ 0 all x∈Ω ⇒ I(f) ≥ 0, that is I preserves positivity;
[…]
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- Information
- Introdction to Measure and Probability , pp. 100 - 133Publisher: Cambridge University PressPrint publication year: 1966