All Questions
Tagged with metric-spaces measure-theory
410
questions
3
votes
1
answer
406
views
Is a metric/distance not a measure?
A metric (https://en.wikipedia.org/wiki/Metric_space) or a distance (as in premetric) takes two elements of a set and maps the pair to a real number (or maybe even a complex number). It also might ...
- 1,131
1
vote
0
answers
33
views
Partition a metric space into parts with small measure and diameter
Consider $\Omega = [0,1]$ with Borel $\sigma$-algebra and Lebesgure measure $\mu$. It has the property that for any $n\geq 1$, we can partition $\Omega$ into $n$ parts with small measure and diameter. ...
- 509
1
vote
1
answer
49
views
Are there compact metric spaces with Hausdorff measure equal to 0 or infinty
I am looking for examples of non-empty metric spaces which are compact with Hausdorff dimension $\alpha$, and have either
its $\alpha$-dimensional Hausdorff measure equal to $0$
or
its $\alpha$-...
- 412
7
votes
1
answer
121
views
Upper semicontinuity of sequence of Hausdorff measures
This is exercise 12.2 of Measure theory and integration of Michael Taylor.
Let $(d_j)$ be a sequence of metrics on a compact space $X$ such that there exists $c,C>0$ with
$$ cd_0(x,y) \leq d_j(x,y) ...
- 636
0
votes
0
answers
28
views
Proving that $L(\mu,\nu)$ is a metric on the space of measures with bounded support and total mass of 1
Let $\mu,\nu\in\mathcal{M}^1$ where $\mathcal{M}^1$ denotes the space of Borel regular measures with bounded support and total mass of $1$ and $\phi\in\mathcal{C}_b$. Let $\mu\colon\mathcal{C}_b\to[0,\...
- 67
0
votes
0
answers
48
views
Integral of ratio of measure of balls
I'm trying to find assumptions that make the equation hold,
$$
\lim_{\epsilon \to 0} \int_\epsilon^\infty \frac{\mu(B(x,t))}{\mu(B(x,\epsilon))} dt = 0,
$$
where $x$ is in a doubling metric measure ...
- 53
2
votes
1
answer
57
views
Proof or Citation that standard Borel spaces contain singletons
Suppose $(\mathsf{X}, \mathcal{B}(\mathsf{X}))$ is a standard Borel space. I have read that then all singletons $\{x\}$ with $x\in\mathsf{X}$ are in $\mathcal{B}(\mathsf{X})$. I cannot find a proper ...
- 159
1
vote
0
answers
21
views
For a Borel subset $B$ of a complete, seperable metric space $S$ and $\epsilon > 0$, there exists compact $C \subset S$ with $P(B) < P(C) + \epsilon$.
For my bachelor thesis, I've been studying iterated random functions and a very limited amount of measure theory to understand it rigorously. One thing I could not understand is the following:
Suppose ...
- 184
1
vote
1
answer
60
views
Generation of the topology of a vector topological space by a metric
I am trying to prove that the set of equivalence classes of measurable functions with the metric $\rho(f,g)=\int_{0}^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|} \, d\mu$ is a vector topological space.
By ...
- 113
2
votes
0
answers
42
views
Measurability of an integral operator?!
Is it possible to prove the measurability of the following map
$\Phi_n \colon (C(\mathbb{R}^d), \sigma(\mathcal{C})) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, $n \in \mathbb{N}$, defined by
$$ \...
1
vote
2
answers
152
views
Solving a problem on metric spaces using measure theory
I recently came across the theorem that every uncountable subset of $\mathbb{R}$ contains a limit point of that set. There is a straightforward proof, if we suppose that each point in some uncountable ...
- 3,044
0
votes
1
answer
41
views
If $\mu$ is a measure, the the collection of all $\mu$-measurable sets is a $\sigma$-algebra
I'm studying from "Topics on Analysis in metric spaces" by Ambrosio Luigi and Paolo Tilli. The Theorem $1.1.6$ is the theorem written in the title. The passage that i don't understand is the ...
- 1,453
3
votes
1
answer
77
views
A metric on $L^1$ using measure of set where $f$ is greater than a value
I am working on a problem showing that the following function describes a metric on $L^1$ for any measure space $(X,\mathcal{S},\mu)$ for a measurable set $E$:
Let -- for $f, g: E \to \mathbb{R}$ ...
- 45
3
votes
0
answers
52
views
Is there any use of finite metrics on $\mathbb{R}$ or extended real numbers $\bar{\mathbb{R}}$?
So in my grad level Real Analysis course based on Mesaure Theory and Lebesgue Integration, our professor added a note stating that extended real numbers $\bar{\mathbb{R}}$ can be used with a finite ...
- 121
1
vote
0
answers
44
views
Metric for weak convergence on space of non-negative measures
Let $X$ be a Polish space, let $\mathcal{M}_+(X)$ be the space of non-negative measures on $X$. What is the metric that metrizes topology of weak convergence (i.e. where test functions are bounded ...
- 41