2
votes
2answers
46 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...
1
vote
1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
1
vote
0answers
24 views

Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
4
votes
2answers
77 views

Formally show that the set of continuous functions is not measurable

Let $C(\mathbb{R})=\{ f:\mathbb{R}\to \mathbb{R} \colon \ f \text{ continuous}\}\subseteq \mathbb{R}^{\mathbb{R}} $. How to prove formally that $C(\mathbb{R}) \notin ...
2
votes
1answer
54 views

can a compact set have infinite measure?

Can a compact set have infinite measure? It does not seem to violate the measure axioms. This is not true in the case of Lebesgue measure. So I am also wondering is there any clean cut condition for ...
1
vote
0answers
25 views

Space of measures is weak-* Hausdorff?

If $X$ is a topological space which is hereditarily Lindelöf and completely regular, then the space of finite signed measures on the Borel $\sigma$-algebra, endowed with the weak-* topology, is ...
2
votes
1answer
65 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
0
votes
1answer
47 views

Luzin's theorem, finding a continuous function under a certain condition

Let $X:\mathbb R^2\rightarrow\mathbb R,$ be the map defined by $(x,y)\mapsto y-x.$ Let $h:\mathbb R\rightarrow\mathbb R$ be Borel measurable. Let $\mu$ be a Borel probability measure on $\mathbb ...
1
vote
0answers
36 views

Prove that the density topology is stronger than Euclidean topology.

I'm working through Franklin Tall's paper on the Density Topology. In theorem 2.3, he defines a topology on $\mathbb{R}$ such that if a set $E \subseteq \phi(E)$, then $E$ is open. He uses $\phi(E)$ ...
0
votes
0answers
94 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
5
votes
1answer
104 views

Connection between separable measure spaces and $\sigma$-finite measure spaces

I recently came across a theorem which makes a hypothesis that a certain measure space is separable (the definition can be found here). In order to avoid confusion, I'll add the definition here: We ...
2
votes
0answers
35 views

Is every sigma-algebra generated by some topology?

Well, my question is precisely what the title says: Is every sigma-algebra on a set $X$ generated by some topology on $X$? Actually,I am unable to either prove or create a counterexample, but I have a ...
4
votes
0answers
42 views

Measurability of the set of all continuous functions

Consider $\mathbb{R}^\mathbb{R}$ the set of all real-valued functions of real variable with the product topology. This topology induces the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^\mathbb{R})$. ...
2
votes
2answers
99 views

Is every continuous function measurable?

In non-Hausdorff topology it is standard to define the Borel algebra of a topological space $X$ as the $\sigma$-algebra generated by the open subsets and the compact saturated subsets. Recall that a ...
2
votes
3answers
51 views

Proof of if $g$ is continuous and $g(x) = 0$ almost everywhere on the closed interval $[c,d]$, then $g(x) = 0$?

Suppose that $g$ is some continuous function on $[c,d]$. Now also suppose that $g(x) = 0$ almost everywhere on the closed interval $[c,d]$. We would like to prove that $g(x) = 0$, $\forall x\in ...
3
votes
2answers
181 views

Why does an open interval NOT have measure zero?

I am currently working on a proof that requires me to show that an open ball $B_{\epsilon}(x)$ has nonzero measure. I currently have the following proof in my book: "The closed interval $[a,b]$ is ...
5
votes
1answer
68 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
2
votes
1answer
46 views

Uncountable disjoint union of measure spaces

Let $(a,b)$ be an interval. Let $(A_i, \Sigma_i, \mu_i)$ be a measure space for each $i \in (a,b)$. Is it possible to put a measure space on the disjoint union $$\bigcup_{i \in (a,b)}\{i\}\times ...
1
vote
0answers
60 views

Sets which are open “modulo a nullset”

A set $A$ is said to have property of Baire there exists an open set $U$ such that $A\triangle U$ is meager. So this says that symmetric difference of $A$ and some open set is small (in the sense of ...
0
votes
1answer
49 views

Hausdorff dimension mathces Box-counting dimension

I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an ...
0
votes
1answer
34 views

Proof strategy - Borel $\sigma-$fields

How does one go about proving the following: Every open set $A$ in the topological space $(\mathbb{R}^d,\|\cdot\|)$ (with the norm topology) is the union of all the open balls $B_\epsilon(q)$ whose ...
1
vote
1answer
64 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
32 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
60 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
47 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
40 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
83 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
51 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 ...
1
vote
1answer
68 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
42 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
54 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
38 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
95 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
41 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
20 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
53 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
89 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 ...
2
votes
1answer
59 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
70 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 ...
3
votes
1answer
35 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
134 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
58 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
72 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
46 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
487 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
17 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
98 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
67 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 ...