Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,071 questions
5
votes
2
answers
641
views
Percolation Model and Complex Probabilities
Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$.
I would like to know, if can we ...
- 929
3
votes
2
answers
766
views
Borel vs measure for all Borel measures
Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
- 41.1k
0
votes
2
answers
234
views
disjointlize an arbitrary sequence in a ring?
In a ring R (nonempty class of sets closed under difference and finite union), any sequence (here means a function on natural numbers $\mathbb N$) {$E_i$} in R can be disjointlized to a disjoint ...
- 8,298
9
votes
2
answers
804
views
Partition of R into midpoint convex sets
We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$.
My question is: is it possible to partition $\mathbb{R}$ ...
- 6,224
2
votes
1
answer
2k
views
If a Borelian set has positive measure, does it contain a non empty open set (minus a measure null set)?
Let A be a borelian set with postivie measure. I was asking myself if it is possible to find an open set $B\subseteq A$ such that $B$ is an open set minus a set of null measure...
- 4,736
1
vote
1
answer
1k
views
A question about sigma-ring
This question comes from the 4th line of the proof of Theorem E of Halmos' "Measure Theory", in page 25, which says that C is a sigma-ring. Because this website does not allow new users to link images,...
- 33.3k
3
votes
1
answer
2k
views
Lebesgue measure of a set
Let m the n-dimensional Lebesgue measure on $R^n$.By definition of product measure, on each borelian set E
$m(E)=\inf \left(\sum_{j=1}^\infty m(R_j),\:\: E\subseteq \bigcup R_j , \:\:R_j \text{ ...
- 783
8
votes
1
answer
381
views
Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
- 45k
0
votes
2
answers
7k
views
lebesgue measure and countable sets [closed]
the lebesgue integral $ \int_{[0,1]} 1_{\mathbb{Q}} dm = 0 $ .
and if we integrate the complement $ \int_{[0,1]} 1_{\mathbb{Q}^C} dm =1 $
which is the same as $\int_{[0,1]} dm $
to me this is still ...
- 9,366
2
votes
1
answer
251
views
Help determining the asymptotic behavior of an integral involving rational functions.
Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that
$$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (...
- 11.8k
10
votes
2
answers
763
views
measure theory for regular cardinals
Measure theory is somewhat focused on the cardinal $\aleph_0$: First of all we have the usual $\sigma$-additivity, Polish (separable!) spaces such as $\mathbb{R}^n$, countable sequences and limits, ...
CommunityBot
- 1
3
votes
2
answers
480
views
Dimension of the measurable space $\mathbb{R}^n$
Consider $\mathbb{R}^n$ as measurable space with the Borel algebra. If $\mathbb{R}^n$ and $\mathbb{R}^m$ are isomorphic (in the category of measurable spaces, i.e. there are measurable maps in both ...
- 65.4k
11
votes
4
answers
3k
views
When does a probability measure take all values in the unit interval?
Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we ...
- 69
4
votes
1
answer
822
views
What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?
I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$?
(For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...
- 11.4k
7
votes
1
answer
2k
views
Why is 3 a bad constant in the Vitali covering lemma?
Hi,
recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
- 7,329
3
votes
1
answer
635
views
Non-existence of integral with respect to Poisson Random Measure
Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).
(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims ...
CommunityBot
- 1
6
votes
2
answers
2k
views
Measure between the counting measure and the Lebegue measure
There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is ...
- 65.4k
5
votes
2
answers
6k
views
Difference between Beta Process and Dirichlet process
I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf
The problem is that from the definition it follows that every ...
- 85.6k
2
votes
5
answers
894
views
Models of the reals which have no unmeasurable sets
I recall being told -- at tea, once upon a time -- that there exist models of the real numbers which have no unmeasurable sets. This seems a bit bizarre; since any two models of the reals are ...
- 7,329
2
votes
2
answers
6k
views
Examples of random variables
I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
- 11.4k
3
votes
3
answers
473
views
Lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type f(n)*delta^{n(n-1)/2}
Is there a lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type
f(n) * delta^{n(n-1)/2}? For which f(n)? How can it be proven?
n(n-1)/2 is the number of degrees of ...
CommunityBot
- 1