Measure Decomposition Theorems

July 18, 2008 by hilbertthm90 | 7 Comments

Well, I’ve been mostly posting comments around on other people’s blogs and not really getting around to my own. I’m giving up on NCG for now. It seems that the stuff I already know I’m reading, and I’m skipping the stuff that will take effort to sort through. This seems pointless, especially with Analysis prelims coming up. That’s why things may take a turn in that direction over the next couple of weeks. I do still have at least one major ethical issue I want to sort out, though.

So. What is the most confusing part of measure theory? To me it is the fact that there are tons of ways to decompose your measure. In fact, I usually can’t remember which one is named what, and when to use which one. This post is an attempt to sort out which one is which, and what to look out for when you want to use them.

Jordan Decomposition: Any real measure $\mu$ on a $\sigma$ -algebra can be expressed in terms of two positive measures, called the positive and negative variations ( $\mu^+$ and $\mu^-$ ) by $\mu=\mu^+-\mu^-$ . This allows us to examine the total variation more easily, since $|\mu|=\mu^+ + \mu^-$ . Also, it is quite simple to prove the existence and uniqueness since we can write $\mu^+=\frac{1}{2}(|\mu| + \mu)$ and $\mu^-=\frac{1}{2}(|\mu|-\mu)$ .

Jordan decomposition seems to be used when you can prove something for positive measures and need to extend it to all measures. Since J decomp gets you any measure in terms of positive measures, this eases the process. The other main use is when you invoke the uniqueness along with the next decomp theorem.

Hahn Decomposition: This is different from all the rest. It is not a decomposition of the measure, but of the measure space. It says: Let $\mu$ be a real measure on a $\sigma$ -algebra $\mathfrak{M}$ in a set X. Then there exist sets A and B in $\mathfrak{M}$ such that $X=A\cup B$ , $A\cap B=\emptyset$ and such that the positive and negative variations $\mu^+, \mu^-$ satisfy $\mu^+(E)=\mu(A\cap E)$ and $\mu^-(E)=-\mu(B\cap E)$ , for any $E\in\mathfrak{M}$ .

Things to note. This is not unique! Also, you get as a quick corollary that since the positive and negative variations are concentrated on disjoint sets, they are mutually singular. The Hahn decomp is usually invoked in conjunction with the J decomp, as in, “Let ____ be the J decomposition and A, B be the respective Hahn decomp.” These two together get you that the J decomp is minimal. In other words, if $\mu=\lambda_1 - \lambda_2$ , where $\lambda_1$ and $\lambda_2$ are positive measures, then $\lambda_1\geq \mu^+$ and $\lambda_2\geq\mu^-$ .

Lebesgue Decomposition: Let $\mu$ be a positive $\sigma$ -finite measure and let $\lambda$ be a complex measure on the same sigma algebra, then there is a unique pair of complex measures $\lambda_a$ and $\lambda_s$ such that $\lambda=\lambda_a + \lambda_s$ and $\lambda_a \ll \mu$ and $\lambda_s \perp \mu$ . Also, if $\lambda$ is positive and finite, then so are the two parts of the decomp. Caution: the measure $\mu$ MUST be sigma finite. This theorem says that given any complex measure and any sigma finite measure, you can decompose the complex one into two unique parts that are absolutely continuous with respect to and mutually singular with the sigma finite measure respectively.

The major use of this is when you want to invoke the Radon-Nikodym theorem to get an integral representation of your measure. The Radon-Nikodym theorem only works if your measure is absolutely continuous with respect to the other. Luckily, with Lebesgue decomposition you can always apply R-D to at least a part of the measure.

Polar Decomposition: Let $\mu$ be a complex measure on a sigma algebra. Then there is a measurable function h such that $|h(x)|=1$ for all x and such that $d\mu=hd|\mu|$ . Note that the name “polar” is in reference to the polar form of writing a complex number as the product of its absolutely value and a number of absolute value 1. I’m not entirely sure I’ve ever used this. I guess the main place that it seems useful is when working with the integral representation of the measure. If you need to manipulate with the total variation, then this gives you how to put it into the integral representation.

Those seem to be the big ones. This is quite possibly the most useful math post I’ve made. I didn’t go into too much depth, but hopefully if someone is struggling with the differences between these, or trying to get a vague idea of when to use them, this post will help. I suppose I could have elaborated a little by proving the simple claims and showing counterexamples for the “cautions.” This would have given a feel for using them. Oh well.

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Author: hilbertthm90

I am a mathematics graduate student fascinated in how all my interests fit together.

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Measure Decomposition Theorems

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