Intereting Posts

regularization of a divergent integral
Square Fibonacci numbers
How to solve the differential equation $y' = \frac{x+y}{x-y}$?
Why this definition of integrability by sup and inf integrals is necessary?
Define the image of the function $f :\Bbb Z \times \Bbb N →\Bbb R$ given by $f(a, b) = \frac{a−4}{7b}$?
Anti-compact space
Help evaluating the integral $\int_{0}^{\infty} \omega \cos(\omega t) \coth(\alpha \omega) \text{d} \omega$.
When each prime ideal is maximal
Reference for Topological Groups
Find all integers $x$ such that $x^2+3x+24$ is a perfect square.
Is this the general solution of finding the two original squares that add up to a given integer N?
The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer
Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?
If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?
Show that the LU decomposition of matrices of the form $\begin{bmatrix}0& x\\0 & y\end{bmatrix}$ is not unique

Prompted by this question I was looking for $A \subset (0,1)$ such that for any interval $(a,b)\subset (0,1), A \cap (a,b)$ and $A^c \cap (a,b)$ are both uncountable. One such $A$ is the set of all numbers that have a finite number of $1$’s in their base $3$ expansion. As no choice was used in the construction, it should be Lebesgue measurable, but I can’t prove it. How is it proved?

- A question about regularity of Borel measures
- Can I understand Egorov's theorem in this way?
- From $\sigma$-algebra on product space to $\sigma$-algebra on component space
- Nonatomic vs. Continuous Measures
- Proving functions are in $L_1(\mu)$.
- Prove Property of Doubling Measure on $\mathbb{R}$
- Positive part of the kernel
- What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve
- Lebesgue integral question concerning orders of limit and integration
- Properties of Haar measure

For each $n$, let $A_n$ be the subset of elements of $A$ that have at least a $1$ in the first $n$ digits of their ternary expansion, but no $1$s after the $n$th digit. Your set $A$ is equal to the union of the $A_n$s and the Cantor set. Each $A_n$ is a finite union of scaled translates of the Cantor set (take $3^{-n}C$, where $C$ is the Cantor set, to get all numbers that have ternary expansion with no $1$s and that have $0$ in the first $n$ positions; then you can “translate” by adding a suitable number that has a tail of $0$s and appropriate $1$s in the appropriate coordinates).

So each $A_n$ is a finite union of Lebesgue-measurable sets (scaled Cantor sets are Lebesgue-measurable, and translates of a Lebesgue-measurable set are Lebesgue-measurable), hence Lebesgue measurable. $A$ is a countable union of Lebesgue measurable sets, hence Lebesgue measurable.

**Added.** As Andres Caicedo points out in the comments below, the argument above shows that the set $A$ is in fact not merely Lebesgue measurable, but *Borel*.

The number $x_n$ in the $n^\text{th}$ place after the radix point in the ternary expansion of $x$ is a measurable function of $x$. This is true because the floor function is Borel measurable, and $x_n=\left\lfloor 3\cdot\left(3^{n-1}x-\lfloor3^{n-1}x\rfloor\right)\right\rfloor$. The set $\{x:x_n\neq 1\}$ is measurable for each $n$, and so therefore is the set $$\bigcup_{n=1}^\infty\bigcap_{k=n}^\infty \{x:x_k\neq 1\}.$$

Certain null sets of the second category give further examples of this. The complement of such a set can be contructed by taking a countable union of closed sets with empty interior whose complements have progressively smaller measure (e.g., using fat Cantor sets).

- What is the difference between matrix theory and linear algebra?
- How do you proof that the simply periodic continuous fraction is palindromic for the square root of positive primes?
- Rings whose spectrum is Hausdorff
- Number of bit strings with 3 consecutive zeros or 4 consecutive 1s
- Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?
- Can we found mathematics without evaluation or membership?
- Mathematical Games suitable for undergraduates
- Sentences in first order logic for graphs
- Non-Standard analysis and infinitesimal
- How do I show that for linearly independent set in dual is a dual of a linearly independent set?
- Prove sum of $\cos(\pi/11)+\cos(3\pi/11)+…+\cos(9\pi/11)=1/2$ using Euler's formula
- Constructing a finite field
- Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.
- Multiplying three factorials with three binomials in polynomial identity
- Nonsingular projective variety of degree $d$