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.
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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)
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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
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