How deep Sets affect Measure Theory?

In summary, the conversation is about real analysis and measure theory, with a focus on the use of set concepts in proofs. The person is addicted to using these concepts and wants to know how often they are used in measure theory. It is noted that sets are always used in measure theory, but topological arguments may vary in frequency.
  • #1
zli034
107
0
Guys,

I'm taking real analysis starting with open, close, compact sets, and neighborhoods. Now I'm addict to rely on these concepts to do my proofs. In the future I will have to take Measure Theory. Can anyone give me a percentage indication for how many percent theorems are proven by the set concepts?

I'm really happy with sets.

zli034
 
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  • #2
Measure theory is really set theory at its core.
 
  • #3
zli034 said:
Guys,

I'm taking real analysis starting with open, close, compact sets, and neighborhoods. Now I'm addict to rely on these concepts to do my proofs. In the future I will have to take Measure Theory. Can anyone give me a percentage indication for how many percent theorems are proven by the set concepts?

I'm really happy with sets.

zli034

If you mean how often are sets used in measure theory: 100%
If you mean how often are topological arguments (such as closed, open, compact) used: it depends, but perhaps 50%
 

1. How do deep sets affect the concept of measure theory?

Deep sets can greatly impact the fundamental concepts of measure theory, as they introduce a new level of complexity and depth to the traditional notions of sets and measures.

2. What is the definition of a deep set in measure theory?

A deep set is a set of points that have a complex internal structure, making it difficult to assign a simple measure to them. It is often described as a set whose elements do not follow a clear pattern or have a well-defined boundary.

3. How do deep sets challenge traditional measure theory?

Deep sets challenge traditional measure theory by introducing the concept of non-measurable sets, which cannot have a well-defined measure assigned to them. This goes against the traditional belief that every set can have a measure assigned to it.

4. What are some applications of deep sets in measure theory?

Deep sets have various applications in fields such as computer science, statistics, and physics. For example, they can be used to analyze complex data sets, improve statistical models, and understand the behavior of quantum systems.

5. How do mathematicians deal with deep sets in measure theory?

Mathematicians have developed new theories and techniques to deal with deep sets in measure theory, such as the theory of non-measurable sets and non-standard analysis. They also use advanced mathematical tools, such as fractal geometry and topological methods, to study and understand these complex sets.

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