# Show that E and the closure of E have the same Jordan outer measure

• cragar
In summary, to prove that E and the closure of E have the same Jordan outer measure, we can use the fact that the Jordan outer measure is monotone. This means that if A is a subset of B, then the measure of A is less than or equal to the measure of B. By assuming that E and the closure of E have different Jordan outer measures, we can find a closed rectangle R that contains E and has a smaller measure than the closure of E. However, using the monotonicity of the Jordan outer measure, we can show that R must have a measure greater than or equal to the closure of E, leading to a contradiction. Thus, we can conclude that E and the closure of E have the same
cragar

## Homework Statement

Let E be a bounded set in $R^n$
Show that E and the closure of E have the same Jordan outer measure

## Homework Equations

Jordan outer measure is defined as $m^* J(E)=inf(m(b))$
where $B \supset E$ B is elementary.
3. The Attempt at a Solution [/B]
If E and the closure of E are the same there is nothing to prove.
other case cl(E) contains limit points of E that are not in E.
Let's first look at the measure of E. BY the definition of Jordan outer measure, The closure of E will contain the inf(B) and E is a subset of B and the closure. Let B, be the closure of E, The inf(B) will be the measure of E.
Now this is where I am a little stuck. The Close of E, has all its limit points, so the inf(E) is part of E, its like a closed interval. There is no set B such that inf(m(B)). Unless i pick B to be the set such that
$B=E+ \epsilon$ so the inf of the measure of B is E.

To prove that E and the closure of E have the same Jordan outer measure, we can use the fact that the Jordan outer measure is monotone, meaning that if A is a subset of B, then m^*(A) <= m^*(B).

Let's assume that E and the closure of E have different Jordan outer measures, meaning that m^*(E) < m^*(cl(E)). Since E is a bounded set, we can find a closed rectangle R such that E is contained in R and m^*(R) < m^*(cl(E)).

Since E is bounded, R is also a bounded set. This means that R is a subset of some elementary set B, which we can write as B = R + ε, where ε is a positive number.

By the monotonicity of the Jordan outer measure, we have m^*(R) <= m^*(B). But since B contains E, we also have m^*(B) <= m^*(cl(E)).

Combining these two inequalities, we get m^*(R) <= m^*(cl(E)). But this contradicts our assumption that m^*(R) < m^*(cl(E)). Therefore, our assumption must be wrong and we can conclude that m^*(E) = m^*(cl(E)).

Hence, E and the closure of E have the same Jordan outer measure.

## 1. What is the definition of Jordan outer measure?

Jordan outer measure is a measure of the size or extent of a set in terms of its boundary points. It is defined as the infimum of the sum of the volumes of all possible coverings of the set by closed rectangles.

## 2. What is the closure of a set?

The closure of a set is the set itself together with all its limit points. It can also be defined as the smallest closed set that contains the original set.

## 3. How is the Jordan outer measure related to the closure of a set?

The Jordan outer measure of a set E is equal to the Jordan outer measure of its closure, denoted as J*(E) = J*(cl(E)). This means that the measure of the boundary points of a set is the same as the measure of the boundary points of its closure.

## 4. Why is it important to show that E and its closure have the same Jordan outer measure?

This is important because it allows us to extend the definition of Jordan outer measure to include all sets, even those that are not bounded or have a well-defined boundary. It also helps us to analyze the size and extent of sets more accurately.

## 5. What are some implications of E and its closure having the same Jordan outer measure?

One implication is that a set E is Jordan measurable if and only if its closure is Jordan measurable. Additionally, if a set has a finite Jordan outer measure, then its closure also has a finite Jordan outer measure. This can also be used in proving the Jordan-Schonflies theorem and other important results in measure theory.

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