All Questions
Tagged with measure-theory real-analysis
551 questions
6
votes
1
answer
6k
views
Change of variables formula for Riemann integration and Lebesgue Integration
I've put this question on math.SE for a while without getting any answers. I thought it must be a rather trivial question for MO so that I didn't put it here. But I do want to get some help anyway (...
user14319
2
votes
2
answers
946
views
Defining definite integral using indefinite integral
Sometimes definite integral is defined using antiderivatives:
$$\int_{a}^b{f(t)dt}=F(b)-F(a)$$
where $F$ is any continuous function such that:
$$(\forall t\in[a,b]\setminus C)(F'(t)\text{ exists and ...
- 63
10
votes
1
answer
1k
views
Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307
2
votes
1
answer
760
views
Apollonian gasket and the degree of convergence
Let $r_1,r_2\dots$ be the radii of Apollonian gasket.
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$
I know that if three circles $A$, $B$ and $C$ ...
- 293
3
votes
1
answer
399
views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\...
12
votes
0
answers
435
views
Uniform closure of subspaces of Baire class 1
Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
- 1,704
1
vote
1
answer
1k
views
Calculating the Lebesgue decomposition of a measure [closed]
How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
- 13
8
votes
3
answers
2k
views
Does a weaker condition than vanishing derivative imply a function being constant?
I learned this question from math.stackexchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does
$$g(x):=\lim_{\epsilon\to 0}\frac{f(x+...
- 183
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
3
votes
1
answer
325
views
Measuring almost-critical values of smooth functions.
Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $x \in X$ is a critical point of $f$ if the gradient $\...
- 15.5k
26
votes
2
answers
12k
views
About the definition of Borel and Radon measures
I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...
- 281
1
vote
3
answers
188
views
sequences of plane measures converging to a singular one: terminology, etc
We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let $a$, $b$, and $c$ be vertices of a triangle in $\mathbb{R}^2$, and $...
- 14k
31
votes
4
answers
8k
views
Counterexamples to differentiation under integral sign?
I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...
- 313
4
votes
2
answers
957
views
Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
- 619
8
votes
2
answers
3k
views
Finite measure on the power set
Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...
- 1,199
32
votes
4
answers
4k
views
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,200
11
votes
4
answers
4k
views
When is the infimum of an arbitrary family of measurable functions also measurable?
Let $(X,\Sigma,\mu)$ be a measure space and consider a family of $\mu$-measurable functions $f_i:X \to \mathbb{R}$ for $i$ lying in some index set $I$. Define $$f(x) = \inf_{i \in I} f_i(x)$$
I think ...
- 15.5k
0
votes
1
answer
721
views
Pointwise limit at Lebesgue's point
Dear MOs,
I am sorry if this problem is too elementary for someone. I just want to get confirmation.
Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
- 1,649
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 161
0
votes
1
answer
857
views
Is Jordan outer measure finitely additive on positively separated sets in $\mathbb{R^n}$?
I am trying to argue that exterior measure has nice properties that Jordan outer measure doesn't have. One of them is finite additivity, but I can't find a simple way to show Jordan outer measure is ...
2
votes
1
answer
4k
views
Uniform $L_1$ convergence implies uniform convergence pointwise a.e.
Let $\Omega$ be a measure space (which can be assumed to be an interval with Lebesgue measure).
It is well known that for a sequence $(f_n)$ in $L^1(\Omega)$ which converges to zero (in $L^1(\Omega)$,...
- 2,270
4
votes
0
answers
213
views
The ring generated by measures
Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
- 8,659
0
votes
0
answers
700
views
Sigma algebra generated
Let $\mathcal{L} \subset \mathbb{R}$ the Lebesgue sigma algebra and $\mathcal{B} \subset \mathbb{R}^{n}$ the Borel sigma algebra. I'll denotes by $\mathcal{L} \times \mathcal{B}$ the smallest sigma ...
- 11
1
vote
1
answer
334
views
Property Sigma Algebra [closed]
Is the set { $ \cup_{i \in \mathbb{N}} C_{i} \times D_{i} : C_{i} \in \mathcal{L} \ , D_{i} \in \mathcal{B}^{n} \ $ } a sigma algebra on $\mathbb{R} \times \mathbb{R}^{n}$ ?
- 11
10
votes
2
answers
3k
views
Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 399
11
votes
2
answers
2k
views
Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 12.8k
0
votes
1
answer
1k
views
surjective function from non-measurable sets
let $V$ be the vitali set and let $g:V\to\mathbb R$ be a surjective function. then the fuction $f:\mathbb R\to\mathbb R$ such that $f(x)=g([x])$ will be a function that is surjective in any interval ...
- 188
0
votes
1
answer
606
views
Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]
Is there a difference between
$L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ?
Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue
$\sigma$-algebra $\...
user20515
3
votes
1
answer
500
views
Hausdorff measure on product spaces of p-adic integers
This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
- 1,723
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
9
votes
2
answers
1k
views
Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
- 1,723
2
votes
0
answers
495
views
Characterization of weak Lebesgue spaces [closed]
I would be interested to know whether the following is true:
Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a ...
- 373
11
votes
4
answers
5k
views
The metric space associated to a measure space
Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \...
- 3,830
8
votes
2
answers
753
views
Patching together homeomorphisms: how badly can it fail?
Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...
- 3,910
2
votes
1
answer
2k
views
Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
- 2,270
3
votes
2
answers
949
views
Reference for proof that $C_b^* = rba$
The following theorem seems to have folk status:
The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, ...
- 459
1
vote
1
answer
1k
views
How to verify the weak convergence?
Given a finite measure on a compact, take $f_n\in L^1$ with norms $\leq 1$ and suppose that $\int f_n g$ tends to a limit for all continuous $g$. Is it true that then $\int f_n g$ converge for any $g\...
- 11
0
votes
1
answer
1k
views
A question about regular signed or complex Borel measure under LRN decomposition
Suppose $\nu$ is a regular signed or complex Borel measure on $\mathbb R^n$, m is the Lebesgue measure on the class of Borel sets $\mathcal B_{\mathbb R^n}$ and the Lebesgue-Radon-Nikodym ...
- 764
7
votes
2
answers
2k
views
Baire Category Theorem Application
In Antoine Henrot Michel Pierre -
Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
- 2,222
21
votes
2
answers
924
views
Codimension of Measurable Sets
I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...
- 8,493
5
votes
0
answers
369
views
Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
- 4,952
9
votes
3
answers
934
views
local behavior of a finite Borel measure
Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall,...
- 1,839
9
votes
2
answers
616
views
construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
- 1,839
26
votes
3
answers
7k
views
Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 29.8k
4
votes
2
answers
767
views
Possible subsets of reals that equal the set of continuity of a function
This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of this past question, though it was motivated mainly by ...
- 7,320
10
votes
4
answers
3k
views
Measure 0 sets on the line with Hausdorff dimension 1
I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $\...
- 505
87
votes
8
answers
16k
views
Why is Lebesgue integration taught using positive and negative parts of functions?
Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
- 50.6k
9
votes
3
answers
4k
views
Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?
A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...
user2529
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 $\mathbb{R}$ ...
- 1,359
9
votes
1
answer
1k
views
Why is this generality in Vitali's Lemma useful?
In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?
Vitali's Lemma:
...
- 466