0
votes
1answer
29 views

Problem with topological space in probability theory. [on hold]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
4
votes
1answer
43 views

Can measure induce a topology on a Set?

When I was taught metric spaces in Topology, I came across the idea that metric defined on a set can induce a topology by creating a basis (open balls). If we have a measure defined on a set, can it ...
3
votes
0answers
59 views

Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
0
votes
2answers
29 views

Topology vs Borel sigma-algebra on a set X

What is the difference between: (X: a set) Topology (open set system) on X Borel sigma-algebra on X Both are a set of open subsets. Both include X and empty set. Both are Closed under union and ...
1
vote
1answer
18 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
1
vote
0answers
19 views

Souslin space and functional

I have a question about Borel $\sigma$-algebra on a Souslin space. Let $E$ be a locally convex topological real vector space which is a Souslin space, that is, the continuous image of separable ...
0
votes
1answer
48 views

Measurability of a pointwise limit of measurable functions

Fellows. I'm trying to prove some measurability result and I figured out a solution using the following and now I wonder if this is actually true. Let $X$ be a topological space and $Y$ be a Polish ...
1
vote
0answers
44 views

Why are empty measurable spaces and empty topological spaces not desirable?

The definition of a $\sigma$-field $\mathscr{F}$ on a set $X$ (or $\sigma$-ring) requires $\mathscr{F}$ to be a non-empty subset of $\mathscr{P}(X)$. Why is this convention taken? What is the issue ...
1
vote
0answers
58 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
2
votes
0answers
37 views

Riesz-Markov-Kakutani Theorem: Various Versions

The Riesz-Markov-Kakutani theorem usually comes in various versions. So I'm a little bit confused and wondering which of these are right. Let $\Omega$ be a locally compact space. Then: Complex ...
1
vote
0answers
62 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
1
vote
1answer
80 views

Is true the boundary of compact set of $\mathbb{R}^n$ have Measure Zero?

Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \rightarrow [0, \infty[$ a measurable function. Suppose that there exist $C>0$ such that $$\int_K f dm < C,\ \forall\ K\subset\Omega,\ K\ ...
1
vote
2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
3
votes
3answers
77 views

non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
0
votes
0answers
28 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
17
votes
1answer
437 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
2
votes
1answer
85 views

Splitting of Nonmeasurable Sets

Being curious I'm wondering: Let $V$ be a Vitali set defined as usually as a choice of $v\in[r]$ with $0\leq v\leq 1$ for every $[r]\in\mathbb{R}/\mathbb{Q}$. Since the countable disjoint union of ...
1
vote
1answer
44 views

The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers. We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under ...
1
vote
0answers
48 views

Classifying topological spaces by measures

While looking at some spaces, I happend to know, that in some spaces (like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)! some examples: in $\mathbb R^n$ ...
1
vote
0answers
37 views

When is an outer Borel regular measure (inner and outer) regular?

Let $X$ be a topological space and $\mu$ an "outer" Borel regular measure on $X$ (for all $A\subset X$, there is $B$ Borel with $\mu(A)=\mu(B)$). Assume that $X=\cup _{i=1}^\infty U_i$, where each ...
3
votes
0answers
22 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
1
vote
0answers
20 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
2
votes
2answers
79 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
24 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
27 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 ...
5
votes
2answers
96 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 ...
3
votes
1answer
65 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
28 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
68 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
37 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
114 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
133 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
37 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
45 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 $\mathcal{B}(\mathbb{R}^\mathbb{R})$. ...
3
votes
2answers
110 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
216 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
70 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
52 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
64 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
60 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
38 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
66 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
33 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
63 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
50 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
41 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
88 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
52 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 ...