Elementary Measure Theory Question

In summary, the conversation is about a question on introductory measure theory. The claim states that if B is the intersection of the rational numbers and the interval [0,1], and \{I_k\}_{k=1}^n is a finite open cover for B, then the sum of the measures of the intervals in the cover is greater than or equal to 1. The proof starts by letting B be a sequence of rational numbers, and then showing that there must be at least one interval in the cover that contains infinitely many elements of B. The author asks for guidance on how to proceed with the proof, as it is currently not rigorous enough.
  • #1
ntsivanidis
4
0
Hey guys, below is a small question from introductory measure theory. Maybe be completely wrong on this, so if you could point me in the right direction I'd really appreciate it.

Claim: Let [tex]B=\mathbb{Q} \cap [0,1][/tex] and [tex]\{I_k\}_{k=1}^n[/tex] be a finite open cover for [tex]B[/tex]. Then [tex]\sum_{k=1}^n m^*(I_k) \geq 1[/tex]

Proof: Let [tex]\ B = \{q_k\}_{k=1}^\infty[/tex].Since [tex]{I_k\}_{k=1}^n[/tex] is a finite cover, there must be at least one [tex] j \in \{1,\dots,n\}[/tex] s.t. [tex]I_j[/tex] contains infinitely many elements of [tex]B[/tex].Fix [tex]\varepsilon > 0[/tex]. WLOG, WMA [tex]I_k=(q_k - \frac{\varepsilon}{2(n-1)}, q_k + \frac{\varepsilon}{2(n-1)}) \ni q_k \ \forall \ k \neq j[/tex]. Then [tex]\sum_{k=1}^n m^*(I_k) = \sum_{k\neq j} m^*(I_k) + m^*(I_j)=\varepsilon + m^*(I_j) \geq m^*(I_j)=1[/tex] since [tex]m^*([0,1]\backslash \mathbb{Q})=1 \ \Box[/tex]
 

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  • #2
Although the statement is correct, the 'WLOG WMA' part of the proof is too sweepy. You must provide details if the proof is to sound rigorous.
 
  • #3
Thanks! Along what lines should I proceed? It's very shaky from that point onwards and I'm stumped for something more concrete. If you could describe a sketch, that would be great. Cheers
 

Related to Elementary Measure Theory Question

1. What is Elementary Measure Theory Question?

Elementary Measure Theory Question is a branch of mathematics that deals with the foundations of measurement and integration. It is concerned with the theory behind measures, which are mathematical functions that assign values to sets.

2. What are the key concepts in Elementary Measure Theory?

The key concepts in Elementary Measure Theory include measures, measurable sets, integration, and measurable functions. Measures are mathematical functions that assign values to sets, measurable sets are sets that can be assigned a measure, integration is a method of calculating the area under a curve, and measurable functions are functions that preserve measurable sets.

3. How is Elementary Measure Theory used in real life?

Elementary Measure Theory has various applications in fields such as physics, economics, and engineering. It is used to model and analyze real-world systems and phenomena, such as stock market trends, population growth, and fluid mechanics.

4. What are some important theorems in Elementary Measure Theory?

Some important theorems in Elementary Measure Theory include the Carathéodory's extension theorem, the Lebesgue differentiation theorem, and the Radon-Nikodym theorem. These theorems provide powerful tools for extending measures, characterizing integrable functions, and decomposing measures, respectively.

5. What are the differences between Elementary Measure Theory and Lebesgue Measure Theory?

Elementary Measure Theory and Lebesgue Measure Theory are closely related branches of mathematics, but there are some key differences. Elementary Measure Theory deals with the basic concepts and properties of measures, while Lebesgue Measure Theory extends these concepts to more complex spaces and functions. Additionally, Lebesgue Measure Theory has a more rigorous and abstract approach, while Elementary Measure Theory focuses on more concrete examples and applications.

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