1
vote
1answer
41 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
1
vote
1answer
25 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
-1
votes
1answer
53 views

Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
0
votes
2answers
41 views

What is the meaning of Common Support here

I am reading a notes in statistical inference, and I am constantly being confused about the term 'common support', i hardly find any definition of this,here is an example, 'Suppose S is a space of ...
1
vote
0answers
30 views

Why is convergence in measure topologizable?

I'm aware that pointwise convergence and uniform convergence are topologizable since the former can be made by seminorms and the latter with a norm. I'm also aware that pointwise a.e. fails because ...
3
votes
1answer
65 views

Isometry vs. measure preserving?

Consider functions between two measured metric spaces. What is the relation between an isometry and a function which preserves the measure of subsets? This question arose in my head as I thought ...
4
votes
1answer
49 views

Whether a set is closed or not

Denote by $C_{[0,1]}$ the ternary Cantor set on $[0,1]$. Now consider $[0,1] \setminus C_{[0,1]}$. It contains open intervals. Now define Cantor sets on all these open intervals by simply translating ...
0
votes
1answer
58 views

A dense subalgebra of $C(X)$ that separates points

Any idea how to do this problem: If $X$ is a compact Hausdorff space and $A$ a subalgebra of $C(X)$ , where $C(X)$ is the algebra of all continuous functions, such that $A$ contains the constant ...
0
votes
0answers
35 views

Does every bounded Lebesgue measurable set of non-zero measure contain a boxed compact set?

Does every bounded Lebesgue measurable set $A$ (of nonzero measure) in $\mathbb{R}^N$ contains a compact set of the form $$I_1 \times I_2 \times \dots \times I_N$$ where $I_i$s are finite closed ...
1
vote
0answers
9 views

Is there a theorem relates continuity of x-section&y-section of a function and continuity(measurability) of a function itself?

Let $(X,\tau),(Y,T),(Z,O)$be topological spaces. Let $f:X\times Y\rightarrow Z$ be a function. Let $f_x,f^y$ denote $x,y$-section of $f$ respectively. Let's assume $f_x,f^y$ are continuous for all ...
1
vote
2answers
53 views

Is the function that gives you the measure of the neighborhood Borel?

Let $X$ be a compact metric space (with $\epsilon -$balls $B_{\epsilon }$) and $\mu $ a Borel probability measure. Let $a,\epsilon >0.$ Is the set $\left\{ x\in X:\mu (B_{\epsilon }(x))\geq ...
2
votes
1answer
35 views

Continuous modification of functions with a given property

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ with the following property: For all reals $x$, $\displaystyle\lim_{y \to x} f(y)$ exists. (In particular, note that its possible that ...
2
votes
1answer
90 views

I have a question about Lebesgue measure

Suppose $A$ is Lebesque measurable, show that for each $\epsilon>0$, there exists an open set, $O$, such that $A\subset O$, and $\lambda(O-A)<\epsilon$. (note: $\lambda$ is the Lebesgue measure ...
4
votes
1answer
35 views

Topological Conditions that imply Non-measurability

I gave a short presentation on Baire spaces, and one of the cute results of the theory I showed is that a Vitali set cannot be nowhere dense. This led me to think that a subset of the reals $A$ which ...
0
votes
0answers
19 views

How do i prove that $\Phi(x,y)=xy$ is measurable where $x,y\in\overline{\mathbb{R}}$?

Let $\overline{\mathbb{R}}$ designate the extended real equipped with the order topology. And let's define $0•\infty=0, 0•-\infty=0, -\infty • \infty=-\infty$. Let $\Phi(a,b)=ab, \forall ...
1
vote
1answer
50 views

A statement on a set and its cluster

Let $X$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism. we define the orbit of a point $x$ as $\mathcal{O}(x)=\lbrace f^n(x): n\in\mathbb{Z}\rbrace$.let $\mu$ be the borel measure ...
1
vote
1answer
69 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
1
vote
1answer
49 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
0
votes
1answer
59 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
2
votes
1answer
30 views

Oxtoby Thm 5.4 Bernstein sets

I am reading Measure and Category of Oxtoby. I have a question about Theorem 5.4 added below. I think I understand the construction of Bernstein sets, and also the main line of the Proof. My question ...
2
votes
2answers
110 views

Closed subset of $[0,1]$ without non-empty open subset and Lebesgue measure greater than $0$

I need to find, for given $0<\alpha<1$, closed subset $C \subseteq [0,1]$ that satisfies $\lambda(C)=\alpha$ ($\lambda$ stands for Lebesgue measure) and includes no non-empty open set. It's ...
1
vote
1answer
43 views

Borel $\sigma$-algebra of the product is the product of the Borel $\sigma$-algebras with $\sigma$-compactness

Suppose $X_1$ and $X_2$ are Hausdorff, locally compact, $\sigma$-compact spaces. Clearly the same holds for their product $X=X_1\times X_2$. We know that in general the Borel $\sigma$-algebra of the ...
2
votes
1answer
70 views

There exists no continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f=\chi_{[0,1]}$ almost everywhere [duplicate]

I am trying to solve the same problem on this page. One gave an hint defining an inclusion function. Does someone know what is meant there? Thanks
0
votes
0answers
45 views

Topology generated by a family of maps and a similar question for measure theory

Let us say that $X$ is a set, $f$ from $X$ to some topological space $Y$, and we endow it with the smallest topology for which $f$ is continuous. Is it true that for any $f_1:X \rightarrow V$ with ...
6
votes
3answers
368 views

“Sum” of positive measure set contains an open interval?

So this homework question is in the context of $\mathbb{R}$ only, and we are using Lebesgue measure. The sum $A+B$ is defined to be $A+B=\{a+b|a\in A,b\in B\}$. The question is: If $m(A),m(B)>0$, ...
0
votes
1answer
15 views

Existence of measure under inverse transformation

Suppose there is nonempty compact metric spaces $X$, $Y$ and a continuous surjective transformation $T : X \to Y$. For given finite measure $\nu$ on $(Y,\mathcal{B})$, is a measure $\mu$ on $(X, ...
3
votes
0answers
78 views

Prokhorov theorem in locally compact Hausdorff space?

Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ...
1
vote
0answers
47 views

Construction of Borel set with given Lebesgue density. [duplicate]

Let $\alpha \in (0,1)$. How can I construct a Borel set $A$ such that $$\lim_{r \to 0+ } \frac{m(A \cap [-r,r])}{2r}=\alpha$$? Thanks.
2
votes
1answer
59 views

Separable set, the real [0,1] interval and measure

I am having a hard time understanding exactly what "separable" means, and I am trying to relate that to the measure of the real [0,1] segment (which is 1, right?). My confusion started when studying ...
0
votes
1answer
87 views

How to make a ghost manifold [closed]

How does one mathematically define a manifold that can pass through another manifold? A "ghost" passing through a "wall" type construction. I understand that this may be done by creating a copy of the ...
1
vote
1answer
51 views

Countable covering and disjoint covering

I am curious about the following problems: Let a bounded set $A\subseteq \mathbb{R}$ be a Lebesgue null set (i.e., $\lambda(A)=0$, $\lambda$ being the Lebesgue measure on the real line). Question ...
0
votes
1answer
106 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
3
votes
0answers
49 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
2
votes
2answers
72 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
0
votes
1answer
60 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
0
votes
1answer
31 views

The closure of the complement of $A \subseteq \mathbb{R}^d$ with Lebesgue measure zero is $\mathbb{R}^d$?

I have been working on an excercise in measure theory for a few hours now, and although I have learned a lot, the answer to this problem avoids me. It concerns proving the following assertion: ...
3
votes
0answers
53 views

On the local measurability

I'm trying to solve this exercise: Let $\mathcal{M}$ a $\sigma$-algebra in $\mathbb{R}^n$ containing the $\sigma$-algebra $\mathcal{B}_n$ of Borel subsets of $\mathbb{R}^n$. We say that a set $A ...
3
votes
2answers
123 views

The Cantor Set length is zero, why? Limits

My question is after the construction of the Cantor set which you might want to skip over that part. This question is not philosophical at all it is pertinent to my understanding of measure and space. ...
0
votes
2answers
82 views

Cantor Set: what is the difference between “measure $0$” and “empty interior”?

I'm confused on what a "empty interior" actually means for the Cantor Set. I recognize that the measure of the set is $0$, but how is that different from being a set with empty interior? For me, ...
3
votes
0answers
34 views

How to push a Borel measure down to the base of a covering.

Suppose I have a covering space $E \rightarrow B$ and a Borel measure on $E$ such that each of the Deck transformations is measure preserving. It seems like there should be a way to push the measure ...
2
votes
3answers
327 views

Countably generated $\sigma$-algebra implies separability of $L^p$ spaces

Let $\Sigma = \sigma(\mathcal C)$ be the $\sigma$-algebra generated by the countable collection of sets $\mathcal C \subset \mathcal{P}(X)$. How can I prove that if $\mu$ is a $\sigma$-finite measure ...
1
vote
1answer
80 views

$\sigma$-algebra generated by open sets coincides with $\sigma$-ring generated by open sets.

Under the topic of Metric spaces in my measure theory book I came across this definition that says: "Denote by $B$ the $\sigma$-ring generated by the class of all the open sets of X. The sets of $B$ ...
1
vote
1answer
63 views

$(\alpha, \infty]$ as a union of open sets.

I would like to show that $(\alpha, \infty]$ is a Borel set using the definition of the Borel $\sigma$-algebra. I'm having trouble with including the infinity, because $\bigcup_n (\alpha, n) = ...
1
vote
1answer
47 views

Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
5
votes
1answer
217 views

$\sigma$-algebras and product topology

What can be said about $\sigma(T_1 \otimes T_2)$ and $\sigma(T_1) \otimes \sigma(T_2)$, when $T_i$ are topologies that aren't necessary second countable, and $\otimes$ denotes, at the left, the ...
3
votes
1answer
76 views

Let $(X,\mathcal{M},\mu)$ be a complete measure space. Show $\mathcal{J}:=\{A\in\mathcal{M}|\mu(A^{c})=0\}\cup\{\emptyset\}\}$ is a topology

Let $(X,\mathcal{M},\mu)$ be a complete measure space. Let $\mathcal{J}:=\{A\in\mathcal{M}|\mu(A^{c})=0\}\cup\{\emptyset\}$ 1) Prove that $\mathcal{J}$ is a topology on $X$ Any thoughts on how to ...
3
votes
1answer
104 views

Is every Borel set finitely decomposable into open sets?

Given a topological space $\langle X, T \rangle $, consider the Borel algebra $B$ generated by $T$. The question is, can you write any $b \in B$ using only finitely many open (or closed) sets using ...
1
vote
1answer
58 views

Subspace of a measure space.

Let $(X,\mu)$ be a measure space and let $Y \subset X$ be $\mu$-measurable with $\mu(Y) > 0$. Define an outer measure $\nu$ on $Y$ by putting $\nu(S) = \mu(S)$ for $S \subseteq Y$. I need to show ...
9
votes
3answers
254 views

Existence of a Minimal Cover

I'm well aware that for the sequence $x_n=\frac{1}{n}$, $\text{inf }x_n=0$ but $0 \notin (x_n)$. This made me think about something similar but when we are no longer thinking about existence of a ...
0
votes
0answers
55 views

An exercise on Lebesgue measure.

I'm studying for my exam, and I found this problem: Let $A$ be a borealian set of extended $\mathbb{R}$ . Show that $A$ has Lebesgue measure $0$ iff for all $\epsilon>0$, exist disjoint open ...