Non-Lebesgue Measurable Sets: Understanding Measurement

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In summary, the existence of non-Lebesgue measurable sets suggests that assigning a number to the Lebesgue measure of such a set is not possible. The Lebesgue outer measure, which is defined for all sets in P(R), is also not enough to determine the measure of a non-measurable set. Depending on how the elements are chosen, the outer measure of the set could be any number between 0 and 1. This raises the question of what the Lebesgue outer measure of a non-measurable set would be, and it seems that it would be impossible to accurately determine.
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dimitri151
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In analysis we were shown the existence of non-Lebesgue measurable sets (eg a choice function over the rational equivalence partition of an interval). From the proof it seems that this means you can't assign number to the Lebesgue measure of this set i.e. if you say its measure is zero it's not enough and if you say it's some finite number then it is too much. However the way we learned Lebesgue measure was that Lebesgue measure was Lebesgue outer measure restricted to a certain family if sets. But Lebesgue outer measure is defined for all sets in P(R). So my question is what is the Lebesgue outer measure (what number) of a non-Lebesgue measurable set like the one above?
 
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I see. But if you specify unambiguously how the elements [itex]x\in[0,1][/itex] are chosen then the set E (the non-measurable set) will have a specific outer measure. But then there will be some set A not disjoint from E such that the sum of the outer measure of the part of A in E and the outer measure of the part of A not in E will be greater than the outer measure of A. It's hard to imagine how this could be.
 

1. What are non-Lebesgue measurable sets?

Non-Lebesgue measurable sets are sets that cannot be assigned a Lebesgue measure, which is a way of measuring the size or volume of a set in mathematics. These sets are often considered to be "too irregular" to have a well-defined measure.

2. Why are non-Lebesgue measurable sets important?

These sets are important because they challenge our understanding of measurement and the concept of size in mathematics. They also have applications in areas such as probability and fractal geometry.

3. How are non-Lebesgue measurable sets different from Lebesgue measurable sets?

Lebesgue measurable sets are sets that can be assigned a Lebesgue measure, while non-Lebesgue measurable sets cannot. This means that Lebesgue measurable sets have a well-defined size or volume, while non-Lebesgue measurable sets do not.

4. Can non-Lebesgue measurable sets exist in physical space?

It is debated among mathematicians whether non-Lebesgue measurable sets can exist in physical space, as their existence would challenge our understanding of measurement. Some argue that they may exist in theoretical or abstract spaces, while others argue that they cannot exist in physical space.

5. How do we study and understand non-Lebesgue measurable sets?

We study and understand non-Lebesgue measurable sets through mathematical analysis and abstract reasoning. This involves exploring their properties, behavior, and relationships with other mathematical concepts. It also involves developing new theories and methods to better understand these sets.

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