# Boundary of open set always zero measure?

• jostpuur
In summary, the conversation discusses whether an open set V in n-dimensional Euclidean space with a boundary of Lebesgue measure zero is Jordan-measurable. The answer is confirmed to be yes, with a reference to Spivak's calculus on manifolds. However, examples are provided where an open set with a positive Lebesgue measure boundary can still be non-Jordan-measurable.
jostpuur
Is this true?

$$V\subset\mathbb{R}^n\;\textrm{open}\quad\implies\quad m_n(\partial V)=0$$

Assuming you mean ordinary Lebesgue measure, the answer is yes.

Are you sure? Do you have a reference?

I have a reference to the countrary! Spivak's calc on manifolds, page 56: "Problem 3-11 shows that even an open set C may not be Jordan-measurable, so that $\int_Cf$ is not necessarily defined even if C is open and f is continuous."

Jordan-measurable means that the boundary has Lebesgue measure zero. And the set of problem 3-11 is A subset of [0,1] given by a union of open intervals (a_i,b_i) such that each rational number in (0,1) is contained in some (a_i,b_i). Then bd(A) = [0,1]\A and if $\sum (b_i-a_i)<1$, bd(A) does not have measure zero.

Another family of examples can be obtained by letting V be the complement of a fat Cantor set in [0,1].

Thanks for pointing out the ugly fact

I was wondering this myself. I think I have an interesting example:
Let Q intersect [0,1] = {r_1, r_2, ...}. Then, find a countable sequence of intervals centered around each r_n, with the property that the total length of the intervals is less than 1. Then, let I be the union of these intervals.

This gives an open set whose boundary has positive lebesgue measure.

I think.

## 1. What is a boundary of an open set?

The boundary of an open set is the set of points that lie on the edge of the open set. It is the boundary between the open set and its complement.

## 2. Why is the boundary of an open set always of zero measure?

The boundary of an open set is always of zero measure because it does not contain any interior points. This means that the boundary consists entirely of points that are on the edge, but not contained within the open set.

## 3. Does the boundary of an open set have any practical applications?

Yes, the boundary of an open set is often used in mathematical analysis and topology to define and study various mathematical objects, such as functions and geometric shapes.

## 4. Can the boundary of an open set be empty?

Yes, in certain cases the boundary of an open set can be empty. This occurs when the open set is also a closed set, meaning that it contains all of its boundary points.

## 5. How is the boundary of an open set related to the concept of a limit point?

The boundary of an open set is closely related to the concept of a limit point. In fact, all points on the boundary of an open set are also limit points, meaning that they are approached by points within the open set. However, not all limit points are necessarily on the boundary of an open set.

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