Abstract
Chapter 7 provides a summary of the aspects of Lebesgue’s measure theory that are used in the earlier chapters. It is in no sense a rigorous introduction to Lebesgue’s theory. Instead, it simply states the central results and attempts to explain their origin as well as their role in probability theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In view of additivity, it is clear that either μ(∅)=0 or μ(A)=∞ for all \(A\in \mathcal{F}\). Indeed, by additivity, μ(∅)=μ(∅∪∅)=2μ(∅), and therefore μ(∅) is either 0 or ∞. Moreover, if μ(∅)=∞, then μ(A)=μ(A∪∅)=μ(A)+μ(∅)=∞ for all \(A\in \mathcal{F}\).
- 2.
I write A n ↗A when A n ⊆A n+1 for all n≥1 and \(A=\bigcup_{1}^{\infty} A_{n}\). Similarly, A n ↘A means that A n ⊇A n+1 for all n≥1 and \(A=\bigcap_{1}^{\infty} A_{n}\). Obviously, A n ↗A if and only if A n ∁↘A∁.
- 3.
The reader should notice the striking similarity between this definition and the one for continuity in terms of inverse images of open sets.
- 4.
When it causes no ambiguity, I use {F∈Γ} to stand for {ω:F(ω)∈Γ}.
- 5.
In this context, we are thinking of [0,∞] as the compact metric space obtained by mapping [0,1] onto [0,∞] via the map \(t\in[0,1]\longmapsto \tan(\frac{\pi}{2}t)\).
- 6.
In measure theory, the convention which works best is to take 0 ∞=0.
- 7.
Although this theorem is usually attributed Fubini, it seems that Tonelli deserves, but seldom receives, a good deal of credit for it.
- 8.
It is convenient here to identify Ω with the set a mappings ω from \(\mathbb{Z}^{+}\) into {0,1}. Thus, we will use ω(n) to denote the “nth coordinate” of ω.
- 9.
This non-uniqueness is the reason for my use of the article “a” instead of “the” in front of “conditional expectation.”
References
Stroock, D.: Mathematics of Probability. GSM, vol. 149. AMS, Providence (2013)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Stroock, D.W. (2014). A Minimal Introduction to Measure Theory. In: An Introduction to Markov Processes. Graduate Texts in Mathematics, vol 230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40523-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-40523-5_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40522-8
Online ISBN: 978-3-642-40523-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)