Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Point sets and certain classes of sets
- 2 Measures: general properties and extension
- 3 Measurable functions and transformations
- 4 The integral
- 5 Absolute continuity and related topics
- 6 Convergence of measurable functions, Lp-spaces
- 7 Product spaces
- 8 Integrating complex functions, Fourier theory and related topics
- 9 Foundations of probability
- 10 Independence
- 11 Convergence and related topics 223
- 12 Characteristic functions and central limit theorems
- 13 Conditioning
- 14 Martingales
- 15 Basic structure of stochastic processes
- References
- Index
6 - Convergence of measurable functions, Lp-spaces
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Point sets and certain classes of sets
- 2 Measures: general properties and extension
- 3 Measurable functions and transformations
- 4 The integral
- 5 Absolute continuity and related topics
- 6 Convergence of measurable functions, Lp-spaces
- 7 Product spaces
- 8 Integrating complex functions, Fourier theory and related topics
- 9 Foundations of probability
- 10 Independence
- 11 Convergence and related topics 223
- 12 Characteristic functions and central limit theorems
- 13 Conditioning
- 14 Martingales
- 15 Basic structure of stochastic processes
- References
- Index
Summary
Modes of pointwise convergence
Throughout this chapter (X, S, μ) will denote a fixed measure space. Consider a sequence {fn} of functions defined on E ⊂ X and taking values in ℝ*. If f is a function on E (to ℝ*) and fn(x) → f(x) for all x ∈ E, then fn converges pointwise on E to f. If E ∈ S and μ(Ec) = 0 then fn → f (point-wise) a.e. (as in Chapter 4). It is clear that if fn → f, fn → g a.e. then f = g a.e. since the limit is unique where it exists.
If fn is finite-valued on E, and given any ∈ > 0, x ∈ E, there exists N = N(x, ∈) such that |fn(x)−fm(x)| < ∈ for all n, m > N, then fn} is said to be a (pointwise) Cauchy sequence on E. If E ∈ S and μ(Ec) = 0, {fn} is called Cauchy a.e. Since each Cauchy sequence of real numbers has a finite limit, if {fn} is Cauchy on E (or Cauchy a.e.) there is a finite-valued function f such that fn → f on E (or fn → f a.e.).
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- A Basic Course in Measure and ProbabilityTheory for Applications, pp. 118 - 140Publisher: Cambridge University PressPrint publication year: 2014