Proving lim m(Ei) = m(E) in Measure Spaces

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In summary, the problem involves a certain measure space and sets in that space with a specific property. The goal is to prove that the limit of the measure of these sets exists and is equal to the measure of the entire space as n approaches infinity. The solution has been found for the finite version, but there is a potential issue if the measure of the union of these sets is equal to infinity. The solution may still apply if the measure of the union is strictly less than infinity, but the negation of this hypothesis is a stronger condition.
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Homework Statement


Let (X , M, m) be a certain measure space and {En} sets in M with the property:
[tex]\underline{lim}[/tex] (Ek) = [tex]\overline{lim}[/tex](Ek) = E
prove that lim m(Ei) exists and = m(E) as n approaches infinity.


Homework Equations





The Attempt at a Solution


i solved the problem, if it were given that m([tex]\bigcup[/tex]Ei) is stirctly less that infinity, but i don't know how to overcome that problem since that was not given.
any help is appreciated.
thank you.
 
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For the finite version you've proved already, it should be enough to have the hypothesis that [tex]m\left(\bigcup_{j\geq n} E_j\right) < \infty[/tex] for some sufficiently large [tex]n[/tex], because your problem isn't changed if you throw away finitely many of the [tex]E_j[/tex] from the front. But the negation of this hypothesis is that [tex]m\left(\bigcup_{j\geq n} E_j\right) = \infty[/tex] for every [tex]n \in \mathbb{N}[/tex], which is a little stronger. Does that help?
 

Related to Proving lim m(Ei) = m(E) in Measure Spaces

1. What is the definition of a limit in measure spaces?

In measure theory, a limit is defined as the value that a sequence of sets or functions approaches as the index of the sequence increases. In other words, it is the value that a sequence of sets or functions converges to.

2. What is the importance of proving that lim m(Ei) = m(E) in measure spaces?

Proving that lim m(Ei) = m(E) in measure spaces is important because it helps us understand the behavior of measures in a given space. It allows us to make precise statements about the convergence of measures and is crucial in many areas of mathematics, including probability theory and analysis.

3. How do you prove that lim m(Ei) = m(E) in measure spaces?

To prove that lim m(Ei) = m(E) in measure spaces, we need to show that for any given value ε, there exists a positive integer N such that for all i ≥ N, the difference between m(Ei) and m(E) is less than ε. This can be done using the definition of a limit and the properties of measures.

4. Can you provide an example of proving lim m(Ei) = m(E) in measure spaces?

Yes, for example, let Ei = [0, 1/i] and E = [0, 0]. Then, m(Ei) = 1/i and m(E) = 0. By the definition of a limit, for any ε > 0, we can find an N such that for all i ≥ N, |1/i - 0| < ε. Therefore, lim m(Ei) = lim 1/i = 0 = m(E).

5. What are some applications of proving lim m(Ei) = m(E) in measure spaces?

Proving lim m(Ei) = m(E) in measure spaces has many applications in various fields of mathematics. It is used in probability theory to study the convergence of random variables and in analysis to prove the convergence of integrals. It is also essential in understanding the behavior of measures in topological spaces and in the study of fractal geometry.

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