Intereting Posts

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma
Number of point subsets that can be covered by a disk
How to find $\lim_{x \to \infty} /x$?
Is there a geometric realization of Quaternion group?
Equivalence of three properties of a metric space.
Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
Hatcher Problem 2.2.36
“Semidirect product” of graphs?
Is Tolkien's Middle Earth flat?
Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero.
Group of order $3^a\cdot5\cdot11$ has a normal Sylow 3-group
On the generating function of the Fibonacci numbers
Monotone increasing sequence of random variable that converge in probability implies convergence almost surely
Proving that isomorphic ideals are in the same ideal class
Group actions transitive on certain subsets

When studying weak border conditions (in Sobolev Spaces), the usual motivation for the weak meaning of inequalities is that the frontier of most open sets in $\mathbb{R}^n$ has zero (Lebesgue) measure. But is there *any* open set $U\subseteq\mathbb{R}^n$ such that it’s frontier has positive Lebesgue measure? I really can’t think of anything like that.

- A limit of a uniformly convergent sequence of smooth functions
- Geometric series of an operator
- Exchange order of “almost all” quantifiers
- Measurable non-negative function is infinite linear combination of $\chi$-functions
- Examples of measurable and non measurable functions
- Prove that the sequence $(n+2)/(3n^2 - 1)$ converges to the limit $0$
- Continuity and the Axiom of Choice
- Continuous function that attain local extrema every point
- When does almost everywhere convergence imply convergence in measure?
- Dominated convergence theorem with $f_n(x)=\frac{1}{\sqrt{2\pi}}e^{-\sqrt{n}x}\left(1+\frac{x}{\sqrt{n}}\right)^n\chi_{}$

There are a lot of open sets in $\mathbb{R}^n$ whose boundary has positive Lebesgue measure. I wouldn’t be surprised if “most” open sets have boundaries with positive Lebesgue measure, where “most” might be referring to cardinality, or some topological or measure-theoretic size.

However, the open sets you can visualize are far more regular than the average open set, so it’s not easy (if at all possible) to get a good mental picture of an open set whose boundary has positive Lebesgue measure. And the open sets one does analysis on, typically also are quite regular and have nice boundaries.

Examples of open sets whose boundary is not a null set are for example complements of a thick Cantor set in dimension $1$ (products where at least one factor is such in higher dimensions).

Somewhat similar, let $(r_k)_{k \in \mathbb{N}}$ be an enumeration of the points with rational coordinates, and let

$$U = \bigcup_{k\in \mathbb{N}} B_{\varepsilon_k}(r_k)$$

for a sequence $\varepsilon_k \searrow 0$ such that $\sum {\varepsilon_k}^n$ converges. Then you have a dense open set $U$ with finite Lebesgue measure, its boundary is its complement and has infinite Lebesgue measure.

- Sum of primitive roots is congruent to $\mu(p-1)$ using Moebius inversion?
- Expression for the Maurer-Cartan form of a matrix group
- Why is a projection matrix symmetric?
- Finding the last two digits of a number by binomial theorem
- Conditions for integrability
- Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power
- Big list of serious but fun “unusual” books
- Boundedness condition of Minkowski's Theorem
- Error solving “stars and bars” type problem
- Represent Dirac distruibution as a combination of derivatives of continuous functions?
- Why is the differentiation of $e^x$ is $e^x$?
- The Riemann zeta function $\zeta(s)$ has no zeros for $\Re(s)>1$
- Is my proof correct? ($A_n$ is generated by the set of all 3-cycles for $n \geq 3$)
- Showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}$ have no gcd
- Number of permutations of $n$ where no number $i$ is in position $i$