Questions tagged [cantor-set]
For questions concerning the Cantor set, which consists of those real numbers in $ [0,1]$ that remain after repeatedly removing the open middle third of every interval; it contains those numbers which may be written in ternary without using 1. Also, for questions about other topological spaces that are homeomorphic to the Cantor set.
557
questions
3
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42 views
A fat Cantor set essentially containing a positive measure subset and uncountably many of its translates
Edit: In writing out more details, I realized I needed to make a slight modification to the question. I switched all proper containments from the old version of the question to "essential ...
1
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0answers
37 views
Let $K \subset [0,1]$ be the Cantor set. show that $f : [0, 1] \to \mathbb R$… [duplicate]
Let $K \subset [0,1]$ be the Cantor set. show that $f : [0, 1] \to \mathbb R$ given by $$f(x) = \begin{cases} 1, & x \in K \\
0, & x \in [0, 1]\setminus C \end{cases}$$ ...
1
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0answers
31 views
Proving the claim that the Cantor set $C$ is perfect [duplicate]
I am self-learning Real Analysis. Stephen Abbott's book leaves the proof of the fact that the Cantor set $C$ has no isolated points as an exercise. I'd like someone to verify, if my construction is ...
0
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0answers
50 views
Bogachev's example: $C+C = [0,2]$ and $C-C = [-1,1]$ where $C$ is the Cantor set [duplicate]
It is known, that the Cantor set is the subset of points of the segment $[0,1]$ allowing ternary expansion $0.a_1a_2\ldots$, where $a_i=0$ or $2,\; i\in {\mathbb N}$, neither of those values ($0$ or $...
1
vote
1answer
68 views
Cantor space and cantor set [duplicate]
I am trying to show that that the the cantor set is homeomorphic to cantor space and here is the function defined from $C$ cantor set to $\mathcal{C}$ Cantor space, let $a_{n}$ be $0$ or $2$ And for ...
1
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0answers
35 views
Proof that $\sum_{n=1}^{\infty}A_n/3^n$ has the Cantor distribution
Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function, according to Wikipedia.
In this question, the answer says that the distribution of
$\...
9
votes
1answer
118 views
Partitioning a metric space into Cantor sets
A "Cantor set" is a topological space which is homeomorphic to the standard Cantor set $C$. In my answer to the question Another way for partition of perfect set by user 00GB I pointed out ...
4
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2answers
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Prove that $C+C=[0,2]$, where $C$ is the Cantor set.
I'd like someone to verify my sketch proof of the below exercise 3.3.7 from Abbott's, Understanding Analysis. If it's incorrect, could you hint/point at the correct approach to the proof. Thanks!
...
1
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0answers
27 views
Proving $μ_F(E) = \int_E F' dλ + μ_F(E∩K)$
The following is an exercise from Bruckner's Real Analysis:
Let $F$ be the Cantor function. Show that, for every Borel set $E$, $μ_F(E) = \int_E F' dλ + μ_F(E∩K)$, where $K$ is the Cantor ternary set....
1
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0answers
48 views
Why is this function increasing?
$K$ : Cantor set
$\phi$ : $K \to [0,1]$, given by $\sum_{k=1}^{\infty} \dfrac{2\epsilon_k}{3^k} \mapsto \sum_{k=1}^{\infty} \dfrac{\epsilon_k}{2^k} \ (\epsilon_k=0,1)$
I hear that this function is ...
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1answer
32 views
Uncoutable Borel Set of Polish Space Contains Cantor Set: Proof Step
Currently, I am struggling to fully understand the following theorem from Alexander Kechris' book “Classical Descriptive Set Theory”:
The last sentence is unclear to me. Why is the homeomorphic copy ...
1
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1answer
52 views
Continuity of map from Cantor set to $[0,1]$
$K$ : Cantor set
$\phi$ : $K \to [0,1]$, given by $\sum_{k=1}^{\infty} \dfrac{2\epsilon_k}{3^k} \mapsto \sum_{k=1}^{\infty} \dfrac{\epsilon_k}{2^k} \ (\epsilon_k=0,1)$
I want to prove that $\phi$ is ...
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0answers
34 views
Showing function whose derivative is bounded but not (Riemann) integrable
This is how the function is constructed in my notes:
Let $C$ be a Cantor set with $\lambda(C)>0$ and $C = [0,1]\setminus U$ where $U = \bigcup\limits_{k=1}^\infty I_k$ is the disjoint union of ...
0
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1answer
47 views
Consider the set, recursively built, starting from the unit interval and removing the first $\frac{1}{3}$ at each step. Find the similarity dimension. [closed]
My thinking for this question is that it is just a slight variation of the standard Cantor set and will therefore have the same similarity dimension. My logic is that at each new step, the interval ...
0
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1answer
24 views
Integrability of The Indicator of a Perfect Set
I am having trouble solving the following problem:
Let $C$ denote the middle-third cantor set. i.e., $C = \bigcap_{i=1}^{\infty}C_n$ where
\begin{align*}
C_1 &= \left[0, \frac{1}{3}\right] \cup \...
2
votes
0answers
45 views
Variance of the random variable with Cantor distribution
I'm currently looking through Oliver Knill's book Probability Theory and Stochastic Processes and am trying to understand Knill's proof that the variance of a random variable $X$ with the Cantor ...
1
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0answers
40 views
Proposition 20, Chap. 2, in Royden's REAL ANALYSIS, 4th edition: The Cantor-Lebesgue Function
Here is the discussion of the Cantor-Lebesgue function in Sec. 2.7, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition:
We now define the Cantor-Lebesgue function, a ...
1
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1answer
49 views
For $f$ the Cantor function there is no function $g$ satisfying $μ_f (E)= \int_E g \ dλ$
The following is an exercise from Bruckner's Real Analysis:
Let $f$ be the Cantor function, and let $μ_f$ be the associated Lebesgue–Stieltjes measure. Show that there is no function $g$ satisfying $...
2
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1answer
135 views
Construction of a function of a countable set of Cantor sets
The following is an exercise from Bruckner's Real Analysis:
I represent only ideas I have for each part and if I am in a right track I'll start fill the gaps and make them rigorous.
(a) ${\{f<a}\}=...
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0answers
9 views
Lower bound of middle third cantor set by using frostman property
I am trying to calculate the lower bound for Hausdorff dimension by using Frostman lemma, if we can show that the Frostman property exist then it will automatically proven Hausdorff measure is nonzero....
2
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0answers
47 views
Points in fifth middle Cantor set sends to points in fifth middle Cantor set. by a function.
Let $$f(x)= \left\{ \begin{array}{lcc}
1-x & if & x \in [0,1]- \{\frac{1}{10}, \frac{1}{2}\} \\
\\ \frac{1}{2} & if & x=\frac{1}{10} \\
\\ \...
3
votes
1answer
110 views
Necessary and sufficient conditions for a function of a specific type on the complement of the Cantor set
The following is an exercise from Bruckner's Real Analysis:
For (a) right limit and left limit must be equal and equals $f(c_n)$ I think this is necessary and also sufficient.
For (b) because each $...
0
votes
1answer
32 views
Existence of a sequence of intervals intersects every point of fat Cantor?
The following is an exercise from Bruckner's Real Analysis:
Let $C$ be a Cantor set in $[0, 1]$ of measure $α$ ($0 ≤ α<1$). Does there exist a sequence ${\{J_k}\}$ of intervals with $\sum_{k=1}^∞ ...
-1
votes
1answer
47 views
Compactness and Cantor's Criterion [closed]
Cantor's Criterion: If $\left\{C_{k}\right\}$ is a nested sequence of closed bounded, nonempty subsets of $\mathbb{R}^{n}$, then
$$
\bigcap_{k=1}^{\infty} C_{k} \neq \emptyset
$$
Furthermore, if $\lim ...
0
votes
0answers
14 views
How to prove that the cantor function $F_x$ satisfies the property $F_X(u)=\dfrac{F_X(3^nu)}{2^n}$ for $u\in(0,3^{-n})$, $n\in\mathbb N$?
Let $F_x$ be the Cantor function. How to prove that it satisfies the property $F_X(u)=\dfrac{F_X(3^nu)}{2^n}$ for $u\in(0,3^{-n})$, $n\in\mathbb N$?
1
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1answer
30 views
What is the Lebesgue measure of a open interval intersected with a generalized Cantor set with positive Lebesgue measure?
In Folland's book Real Analysis: Modern Techniques and Their Applications, p. 39 has an explanation of how to construct a generalized Cantor set with positive measure.
For reference, the construction ...
0
votes
1answer
44 views
The measure of a Cantor Set [duplicate]
So I am trying to argue that the set of points in $[0,1]$ which will not have $4$ in their decimal expansion has measure $0$. So I am thinking to construct a step function such that this set is a null ...
3
votes
1answer
99 views
If $f(x)=x^3-\lambda x$ then $\Big\{ x\in\mathbb R : \Big| \lim_{n\to \infty}f^n(x)\Big|<+\infty \Big\}$ is a Cantor set.
This problem is from An introduction to chaotic dynamical systems Robert L. Devaney
I've already prove (a) and (b) but i'm a bit confused about (c).
For (c) we ...
1
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2answers
105 views
Show that a number belongs or does not belong to a Cantor set
I have trouble showing that $\frac{10}{31}$ and $\frac{45}{69}$ belong/do not belong to the Cantor set. I tried to do the following
$$\sum_{n = x}^\infty \frac{2}{3^n} = \frac{2}{3^x} + \frac{2}{3^{(x ...
1
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1answer
25 views
Condensation points and Lebesgue-zero measure set
Definition. $x \in \mathbb{R}$ is a condensation point of a subset $A \subseteq \mathbb{R} \iff$ the intersection of every neighbourhood of $x$ with $A$ is uncountable.
Can one construct a subset $N \...
0
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1answer
45 views
Representation of numbers in Cantor set as ternary expansion - in need of a rigorous proof
Background Facts
It is known that every number in the Cantor set $\mathcal{C}$ has a ternary expansion,
where $a_k \in \{0,2\}$:
$$
x=\sum_{k=1}^{\infty} a_{k} 3^{-k}
$$
Cantor set has the ...
0
votes
1answer
28 views
Cantor set and evaluating an integral
I'm trying to prove that the following function is Riemann integrable on $[0,1]$: $$g(x) = \begin{cases} 5, &x\in C \\ x, &x\not \in C\end{cases}$$
where $C$ denotes the Cantor set. It's well ...
1
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0answers
36 views
Compact Zero-Dimensional, Planar Sets are Contained in Planar Cantor Sets?
It's known that a zero-dimensional, compact metric space $X$ embeds in the Cantor Set: Take a countable, clopen basis $(U_j)$ indexed by $\mathbb{N}$ and construct a map $f: X \rightarrow \lbrace 0, 1 ...
0
votes
1answer
21 views
BV[a,b]$\cap$C[a,b]$\neq$AC[a,b]
Take an $f\in$BV[0,1]$\cap$C[0,1] e.g. the Cantor function.
I take the Lebesgue Stiltigies measure of $f$:
$$
\mu_f((a,b])=f(b)-f(a).
$$
Now I have a finite positive measure, so I can do the Radon-...
1
vote
1answer
88 views
Is this a probability measure on the cantor set?
$c$ is the Cantor function, $C$ the Cantor set, and $\rho$ the Lebesgue measure
We consider $\mu$ as $\mu(A) = \rho(c(A \cap C))$ for each each element of the tribute
Is $\mu$ currently defining ...
0
votes
3answers
67 views
Cantor set intersected with $\mathbf{R}\setminus\mathbf{Q}$
I am trying to construct an example of a perfect set that contains no rational.
I see in other places a construction that involves an enueration of rationals, but I thought that construction was a bit ...
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votes
1answer
36 views
Is the graph of $f$, $G_{r}(f)$, is connected?
Consider the Cantor set $C$ in $[0,1]$. And a function $f: [0,1] \rightarrow [-1,1]$ defined by
$$f(x) = \frac{2(x-a)}{b-a}-1 \text{ if } x \in [a,b] \text{ where }(a,b) \text{ is a contiguous ...
0
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0answers
19 views
Prove the set of contiguous interval of Cantor set is countable.
Prove the set of contiguous interval of Cantor set is countable.
Remember that means one of the "holes" you cut out when constructing the Cantor set and we know that Cantor set is ...
0
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0answers
24 views
Prove there is a sequence $\{x_n\}, x_n $ in Cantor set and not interval contiguous such that $x_n \rightarrow \frac{1}{3}$
Let $C=$ Cantor set. We know by construction of Cantor set that endpoints of contiguous intervals of the cantor set, for example $(\frac{1}{3},\frac{2}{3})$ is a contiguous interval and $\frac{1}{3}, \...
0
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0answers
35 views
Properties about Cantor sets.
Let $C$ be the usual Cantor defined $C = \bigcap F_n$ where $F_1 = [0,1]$, $F_2 = [0,\frac{1}{3}]\cup[\frac{2}{3},1]$, $\dots$. I am trying to prove some elementary properties about this set and was ...
6
votes
1answer
93 views
How to show two properties about the Cantor Set
Define $C_0=[0,1]$ and for $n\in\mathbb{N}$, define $$C_n=C_{n-1}\setminus\bigg(\bigcup_{k=0}^{3^{n-1}-1}\bigg(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\bigg)\bigg) $$ Then the Cantor set is defined as $$C=\...
0
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1answer
73 views
Homeomorphic images of endpoints of the Cantor set.
A Cantor set is a homeomorphic image of the standard ternary Cantor set $T$. Suppose that we have a Cantor set $C$ on the plane. It is well know that $C$ is in fact ambiently homeomorphic to $T$, ...
0
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0answers
28 views
continuous map from cantor set to another set
Let $C$ be the Cantor set with metric from $(\mathbb{R},|\cdot|)$ and $P = \prod_{k=1}^\infty \{0,\frac{1}{2^k}\}$ with metric from $(\ell_1, \lVert \cdot \rVert_1).$ Show that there exists a ...
2
votes
3answers
81 views
What does it mean to find something in Set theory? What is formal-language equivalent of “find”?
For example in proof that every real number $x$ has a decimal expression $x=a_0.a_1a_2a_3….$ it says:
so we can find $a_1$ between $0$ and $9$ such that...
https://math.stackexchange.com/a/2625318/...
3
votes
3answers
101 views
If the set of specific sequences of 0's and 1's is countable then it can be used to count the set of all infinite sequences of 0's and 1's. Right?
I am seeking some help to understand why an infinite sequence of $\{0, 1\}$ is uncountable. While there are similar questions here with detailed answers, I wasn't able to resolve the contradiction ...
0
votes
1answer
65 views
How can the Cantor set be uncountable, yet its complement is the union of only countably many disjoint open intervals? [duplicate]
$C$ = Cantor set. $C$ is closed, so $C^c$ is the countable union of disjoint open intervals. So let $$C^c = \bigcup\limits_{\substack{i~\in~\mathbb{N} \\}} A_i $$
where $A_i$ are disjoint open ...
0
votes
1answer
39 views
Where does surjectivity fail in a mapping from the unit interval to the Cantor Set?
I'm asked to state why there cannot be a surjective and continuous function $$g: [0,1]\longrightarrow
\text{Cantor Set} $$
I know that $g^{-1}$ exists and is continuous & surjective since the ...
2
votes
2answers
88 views
Proving $g:P\to\Bbb R, g(x)=3x+2$ is discontinuous
Define $g:P\to\Bbb R,g(x)=3x+2$ where $P$ is the Cantor set. Show that $g$ can't be continuous under Euclidean metric in $\Bbb R$.
No information is given about the metric on $P$. I assumed it is the ...
2
votes
1answer
37 views
Union of deletion Cantor sets
I know that it is possible to have deletion Cantor set which are of non-zero measure (fat Cantor sets) furthermore it must never of measure one since it would result in a contradiction (as Cantor is ...
3
votes
1answer
41 views
Ternary and Binary representations to Prove Cantor set is uncountable (questions)
I am trying to prove that the Cantor set is uncountable, but I am very much a novice in working with different bases of numbers. I've never had to do it in any of my classes until now.
Where I'm at is ...
