Can Measurable Sets Be Written as Disjoint Union of Countable Collection?

In summary, the conversation discusses a lemma used in a proof regarding necessary and sufficient conditions for a measurable set. The lemma states that if a set is measurable and has infinite outer measure, it can be written as a disjoint union of countable measurable sets with finite outer measure. There is some uncertainty about the generality of this lemma and its validity in certain cases.
  • #1
Bashyboy
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Homework Statement


I am working through a theorem on necessary and sufficient conditions for a set to be measurable and came across the following claim used in the proof: Let ##E## be measurable and ##m^*(E) = \infty##. Then ##E## can be written as a disjoint union of a countable collection of measurable sets, each of which have a finite outer measure.

Homework Equations

The Attempt at a Solution



I am not really sure where to begin. I have searched through my book and haven't found any theorem/lemma even remotely like this. Is this lemma used part of more general theorem? If so, what does that theorem look like? I did a google search and couldn't find anything.
 
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  • #2
Bashyboy said:
Let ##E## be measurable and ##m^*(E) = \infty##. Then ##E## can be written as a disjoint union of a countable collection of measurable sets, each of which have a finite outer measure.
Is this even true in full generality? I don't think so: What happens when your measure space consists of a single point ##x## such that ##E = \{x\}## has infinite measure?
 
  • #3
Krylov said:
Is this even true in full generality? I don't think so: What happens when your measure space consists of a single point ##x## such that ##E = \{x\}## has infinite measure?

I am not sure...I have just begun studying measure theory. I sure hope it's true, otherwise I am going to need to contact Royden and Fitzpatrick! I checked my book again, and that is the very lemma they are using in their proof.
 

What is the Theorem About Measurable Sets?

The Theorem About Measurable Sets is a mathematical theorem that states that a set is measurable if and only if its outer measure is equal to its inner measure. In other words, a set is measurable if it can be approximated from both the inside and the outside by a sequence of simpler sets.

Why is the Theorem About Measurable Sets important?

The Theorem About Measurable Sets is important because it provides a rigorous definition of what it means for a set to be measurable. This is crucial in many areas of mathematics, such as measure theory, probability theory, and real analysis. It also allows for the development of important concepts and results, such as the Lebesgue measure and integral.

What are some real-world applications of the Theorem About Measurable Sets?

The Theorem About Measurable Sets has many real-world applications, particularly in fields such as physics, economics, and engineering. For example, it can be used to model the behavior of fluids in motion, analyze economic data, and design efficient algorithms for data processing.

How is the Theorem About Measurable Sets proven?

The Theorem About Measurable Sets is proven using the concept of outer measure, which is a function that assigns a non-negative real number to each set. The proof involves showing that if a set is measurable, then its outer measure is equal to its inner measure. This is done using techniques from real analysis and measure theory, such as the Carathéodory extension theorem.

Are there any limitations to the Theorem About Measurable Sets?

While the Theorem About Measurable Sets is a powerful tool in mathematics, it does have some limitations. For example, it only applies to sets in Euclidean spaces and does not hold for more general metric spaces. Additionally, it does not provide a method for actually determining whether a given set is measurable or not.

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