Construct an open subset of [0,1] of arbitrary measure dense therein

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In summary, the conversation discusses constructing an open subset of [0,1] with Lebesgue measure m(E) = ε and proving its density in [0,1]. The idea of using the fat Cantor set is suggested, and the need for help in proving its density is mentioned. The conversation also mentions the possibility of finding a subset of [0,1] with measure zero that is dense.
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benorin
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Q: Construct an open subset E of [0,1] having Lebesgue measure [itex]m(E)=\epsilon[/itex] such that [itex]0<\epsilon<1[/itex] which is dense [0,1].

A: The fat Cantor set. I need help proving it is dense in [0,1]. The usual Ternary expansion argument stuff won't work as the sets used are of length [tex]\frac{\epsilon}{3^k}[/tex] at the kth iteration. Ideas?

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  • #2
Do you mean proving that [0,1]/C is dense?
 
  • #3
Yes, sorry. The set at large, namely E, is the complement of the so-called fat Cantor set C in [0,1], i.e. [0,1]\C.
 
  • #4
Can you find a (not necessarily open) subset of [0,1] that is dense and has measure zero?
 

What does it mean to construct an open subset of [0,1] of arbitrary measure dense therein?

Constructing an open subset of [0,1] of arbitrary measure dense therein means finding a subset of the interval [0,1] that is open (meaning it does not contain its boundary points) and has a specific measure (or size) that can be any value between 0 and 1. Additionally, this subset must be dense, meaning that it contains points from the entire interval [0,1] and has no gaps.

Why is constructing such a subset important in scientific research?

Constructing an open subset of [0,1] of arbitrary measure dense therein is important in scientific research because it allows for the creation of a mathematical model that can be used to study and understand various phenomena. This type of subset is often used in probability and measure theory, and has applications in fields such as physics, economics, and computer science.

What is the process for constructing such a subset?

The process for constructing an open subset of [0,1] of arbitrary measure dense therein involves using mathematical techniques such as the Vitali covering theorem or the Lebesgue density theorem. These theorems provide methods for constructing a set with specific properties, such as being open and dense, within a given interval. The process may also involve using mathematical tools such as measure theory and topology.

Can this subset be constructed with a specific measure in mind?

Yes, this subset can be constructed with a specific measure in mind. The process for constructing the subset involves choosing a measure (or size) for the subset, and then using mathematical techniques to create a set with that specific measure. This allows for flexibility in the construction process and allows for the subset to be tailored to a specific research question or application.

Are there any limitations to constructing an open subset of [0,1] of arbitrary measure dense therein?

While it is possible to construct such a subset, there are limitations to consider. For example, depending on the chosen measure, it may not be possible to construct a subset that is both open and dense within [0,1]. Additionally, the construction process may become more complex as the desired measure becomes smaller or larger. Overall, the limitations depend on the specific properties and measures desired for the subset.

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