# Proving Finite Outer Measure Inequality

• Artusartos
In summary, the conversation involved finding an open set containing a non-measurable set E with finite measure, where the outer measure of the complement of E in O is greater than the difference between the measure of O and E. The attempt at solving the problem focused on proving the last inequality, but it was pointed out that this does not prove the existence of such an open set. To find the desired set, the definition of "measurable" should be used.
Artusartos

## Homework Statement

Let E have finite outer measure. Show that if E is not measurable, then there is an open set O containing E that has finite measure and for which

m*(O~E) > m*(O) - m*(E)

## The Attempt at a Solution

This is what I did...

$m^*(O) = m^*((O \cap E^c) \cup m^*((O \cap E)) \leq m^*(O \cap E^c) + m^*(E)$

So...

$m^*(O) - m^*(E) \leq m^*(O \cap E^c)$

But I'm confused about the equality, because the one in the question is a strict inequality

Last edited:
Artusartos said:

## Homework Statement

Let E have finite outer measure. Show that if E is not measurable, then there is an open set O containing E that has finite measure and for which

m*(O~E) > m*(O) - m*(E)

## The Attempt at a Solution

This is what I did...

$m^*(O) = m^*((O \cap E^c) \cup m^*((O \cap E)) \leq m^*(O \cap E^c) + m^*(E)$

So...

$m^*(O) - m^*(E) \leq m^*(O \cap E^c)$

But I'm confused about the equality, because the one in the question is a strict inequality
What you have done is just to prove that if there's an open O that contains E, then the last inequality in your post is satisfied. You have done nothing to prove that such a set exists. Of course, the entire space is always an open set, so that's not really an issue, at least not when the space has finite measure. But this still suggests that you need to find a special kind of open set, not an arbitrary one. And you should expect to have to use the definition of "measurable" to find it.

It would have been a good idea to include the definition of "measurable" under "relevant equations".

## 1. What is meant by "finite outer measure inequality"?

Finite outer measure inequality is a mathematical concept that refers to the relationship between the size of a set and its measure. It states that the measure of a set cannot be larger than the sum of the measures of its subsets.

## 2. Why is proving finite outer measure inequality important?

Proving finite outer measure inequality is important because it is a fundamental concept in measure theory, which is a branch of mathematics that deals with the size and properties of sets. This inequality is used to establish the validity of many other theorems and results in measure theory.

## 3. How is finite outer measure inequality proven?

Finite outer measure inequality can be proven using a variety of methods, such as the Carathéodory's criterion or the Vitali covering lemma. These methods involve constructing a sequence of nested sets to approximate the original set and showing that the measure of the original set is smaller than the sum of the measures of the nested sets.

## 4. What are some real-life applications of finite outer measure inequality?

Finite outer measure inequality has many real-world applications, particularly in statistics and probability. It is used to analyze the properties of random processes, such as the convergence of random variables, and to prove the existence of certain statistical distributions.

## 5. Are there any extensions or variations of finite outer measure inequality?

Yes, there are several extensions and variations of finite outer measure inequality, such as the Lebesgue outer measure inequality and the Hausdorff outer measure inequality. These variations have different conditions and assumptions, but they all serve to establish the relationship between the measure of a set and its subsets.

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