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modestoraton
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If i have a measurable set with positive measure, how do I prove that there are 2 elements who's difference is in Q~{0} (aka a rational number that isn't 0.
Proving the Existence of Rational Differences in a Measurable Set
- Thread starter modestoraton
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In summary, measure theory is a branch of mathematics that deals with defining and measuring the size of sets and their subsets. It is important because it provides a consistent and precise way of comparing sizes and serves as a foundation for other branches of mathematics. The main components of measure theory are measures, measurable sets, and integration. Lebesgue measure and Borel measure are two types of measures, with Lebesgue being more general and Borel being specific to certain types of sets. Furthermore, measure theory has real-life applications in various fields such as physics, engineering, economics, and biology.
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Hi modestoraton! 
Let M is measurable such that there are no two elements who's differences are in [itex]\mathbb{Q}\setminus \{0\}[/itex]. Let [itex]M_n=M\cap[n,n+1][/itex].
Then perhaps you could evaluate the sum
[tex]\lambda\left(\bigcup_{q\in\mathbb{Q}\cap[-1,1]}{q+M_n}\right)[/tex]
and show that Mn has measure zero.
Let M is measurable such that there are no two elements who's differences are in [itex]\mathbb{Q}\setminus \{0\}[/itex]. Let [itex]M_n=M\cap[n,n+1][/itex].
Then perhaps you could evaluate the sum
[tex]\lambda\left(\bigcup_{q\in\mathbb{Q}\cap[-1,1]}{q+M_n}\right)[/tex]
and show that Mn has measure zero.
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modestoraton
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Thank you so much.
1. What is measure theory?
Measure theory is a branch of mathematics that deals with the concepts of size, length, and volume of sets and their subsets. It provides a formal framework for defining and measuring the size of sets, which are often referred to as measurable sets.
2. Why is measure theory important?
Measure theory is important because it allows us to define and compare the sizes of sets and their subsets in a precise and consistent manner. It also provides a foundation for various other branches of mathematics such as probability theory, functional analysis, and differential geometry.
3. What are the main components of measure theory?
The main components of measure theory are measures, measurable sets, and integration. A measure is a function that assigns a size to a set, measurable sets are sets whose size can be determined by a measure, and integration is a way of calculating the size of a set by dividing it into smaller, measurable subsets.
4. What is the difference between Lebesgue measure and Borel measure?
Lebesgue measure is a type of measure that is defined on all subsets of a given space and is used to measure the size of sets in a more general sense. Borel measure, on the other hand, is defined only on Borel sets, which are a specific type of measurable sets that are generated from open sets. Borel measure is often used in probability theory and statistics, while Lebesgue measure is used in more general mathematical contexts.
5. How is measure theory applied in real life?
Measure theory has many practical applications in fields such as physics, engineering, economics, and biology. For example, it is used in physics to calculate the volume of irregularly shaped objects, in engineering to determine the length of curves and surfaces, in economics to measure consumer preferences, and in biology to quantify the size of populations and their growth rates.
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