Questions relating to measures, measure spaces, Lebesgue integration and the like.
13
votes
0answers
175 views
Measure theoretic definition of curl
Is there a good measure theoretic definition of curl?
To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet ...
12
votes
0answers
812 views
When is the image of a null set also null?
It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ ...
11
votes
0answers
133 views
pointwise limit of finite measures
If there is a sequence of measures $\mu_n$ such that $\mu_n(A) \overset{n}{\rightarrow} \mu(A)$ for all $A$ in the $\sigma$-field and if $\mu_n(\Omega)\leq c$ $(c<\infty)$ for all $n$, then $\mu$ ...
9
votes
0answers
91 views
Restrictions of null/meager ideal
Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
9
votes
0answers
136 views
Must the Minkowski sum of a Borel set and a *closed* ball be Borel?
Let $A$ be a Borel set in $\mathbb{R}^n$. Must then $A + B(0,1)$ be Borel?
Here $B(0,1)$ is the closed ball centered at $0$ of radius $1$.
I know that Erdos and Stone gave an example of a compact set ...
9
votes
0answers
158 views
Is there a similar concept for a sigma algebra like a base for a topology?
For both a sigma algebra and a topology, we can talk about their generators.
For a topology, a base is a special generator only using union, which is a useful concept in topology.
In parallel ...
9
votes
0answers
201 views
A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals
Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$.
We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
8
votes
0answers
172 views
Generated sigma algebra from Brownian Motion
Suppose that we have a Brownian motion and we define the P-augmented filtration by
$$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$
where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
7
votes
0answers
128 views
What properties are preserved under a measurable mapping?
Although in an abstract category the morphisms are not explicitly defined, in a concrete example (model theory?), morphisms are (always/usually?) mappings that preserve some properties.
In the ...
7
votes
0answers
155 views
A type of local minimum (2)
Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. ...
7
votes
0answers
313 views
Cauchy-Formula for Repeated Lebesgue-Integration
Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given.
Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
6
votes
0answers
106 views
Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$
I am trying to solve a qualifying exam problem.
I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
6
votes
0answers
76 views
Measure-driven differential equations
Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
6
votes
0answers
187 views
Compact set of probability measures
I think I can solve the following exercise if X is assumed to be separable, otherwise I can't.
Let X be a (Hausdorff) locally compact space, $\pi : X \to Y$ a continuous map into a topological space ...
6
votes
0answers
262 views
Existence of non-atomic probability measure
The Question
Let $X$ be a set. Let $\mathcal{F}\subseteq P(X)$ be a $\sigma$-algebra. (Or, if it makes a difference, let $X$ be a topological space and $\mathcal{F}$ the Borel sets.) When can we ...
5
votes
0answers
106 views
+50
Need an explanation of this paragraph “Measure Theory”
I will just quote a part of one proof in "On uniformly regular topological measure spaces by Babiker: page 781" vol43 No4 Duke Math. J. 1976.
Let $I$ be the unit interval endowed with Lebesgue ...
5
votes
0answers
125 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
5
votes
0answers
122 views
Measurability of a certain set in Falcolner's Geometry of Fractal Sets
On page 24 of Falcolner's The Geometry of Fractal Sets, Falcolner defines the set $F = \{ x \in E : \mathcal{H}^s(E \cap U) < \alpha$ diam$(U)^s$, for all convex sets $U$ containing $x$ such that ...
5
votes
0answers
42 views
Regularity of test functions in the definition of the total variation distance
As is well-known, the total variation distance between (the laws of) two random variables $X$ and $Y$ defined on $\mathbb{R}$ is given by
$\sup |E[g(X)]-E[g(Y)]|$, where the supremum is taken over all ...
5
votes
0answers
106 views
Complex measure
Let $\lambda, \mu$ be complex measures on $(X,\alpha)$ and $(Y,\beta).$
Prove there exist a unique complex measure $\lambda\times \mu$ on the sigma algebra $\alpha\otimes \beta,$ such that ...
5
votes
0answers
148 views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in ...
5
votes
0answers
40 views
Shift operator on locally compact groups
Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively).
...
5
votes
0answers
169 views
Portmanteau theorem for vague convergence
I would like to investigate if an analog of the classical Portmanteau theorem holds for vague convergence of Radon measures.
Here are the definitions I'm using.
Let $X$ be a Hausdorff locally ...
5
votes
0answers
85 views
Haar measure existence using distributions
The book of Lieb and Loss proves the existence of Lebesgue measure in an unorthodox way as theorem 6.22, using the fact that ''positive distributions are measures''.
My question is, whether it is ...
5
votes
0answers
155 views
Lipschitz continuity of an integral
Let $(E,d)$ be a metric space, $\mathscr E$ be its Borel $\sigma$-algebra and $\mu$ be a $\sigma$-finite measure on $(E,\mathscr E)$. Let the function $p:E\times E\to\mathbb R_+$ be non-negative and ...
5
votes
0answers
260 views
Rudin's Complex and Real Analysis: Regular measure
This is for anyone who has Rudin's Real and Complex analysis book at hand. I was looking at Rudin's Theorem 2.17 and 2.18. So far everything makes sense, except for one statement that Rudin makes in ...
5
votes
0answers
417 views
Monotone class theorem
I have some question about the Monotone Class Theorem and its application. First I state the Theorem:
Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that ...
4
votes
0answers
67 views
A combinatorial problem.
Let be $(X, \mathbb{A}, \mu)$ a measure space, a partition of $ X $ is a disjoint family $\xi=\{P_1,\ldots,P_k \}$ of measurable sets such tath $\bigcup P_i=X\pmod0).$
If $\xi=\{P_1,\ldots,P_k ...
4
votes
0answers
50 views
Conditional expectation as a random variable
We have three random variables $x,y,z$. Is the condition "$y$ and $z$ are independent" enough to guarantee that "$\mathbb{E}(x\,|\,y)$ and $z$ are independent"? Would anyone give me a brief proof or ...
4
votes
0answers
104 views
strict convexity with a measure theoretic property
Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
4
votes
0answers
64 views
C* algebra of bounded Borel functions
Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
4
votes
0answers
62 views
Baire space homeomorphic to irrationals
I try to show that the Baire space $\Bbb N^{\Bbb N}$, with regular product metric, is homeomorphic to the unit interval of irrationals $(0,1)\setminus\Bbb Q$. I already know that the needed function ...
4
votes
0answers
117 views
Disintegration of Measures
I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background.
Let $(\Omega, ...
4
votes
0answers
146 views
“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?
Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e.
Is there a ...
4
votes
0answers
100 views
Three properties of the Lebesgue measure on $\mathbb{R}^n$
I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$.
It is a non-negative ...
4
votes
0answers
124 views
Uniqueness of Haar Measures
Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
4
votes
0answers
75 views
Find optimal measure
Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem
$$
\int\limits_{\Omega}f(x)\,\mu(dx) \to ...
4
votes
0answers
88 views
What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?
What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$?
As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
4
votes
0answers
80 views
An exercise about the regular Borel measures
I want to prove following (from Folland, Ex. 3.26): If $\lambda$ and $\mu$ are positive, mutually singular Borel measures on $R^n$ and $\lambda + \mu$ is regular, then so are $\lambda$ and $\mu$.
For ...
4
votes
0answers
75 views
A question about functions in $L^p(E)$
I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
4
votes
0answers
81 views
Haar Measure: Unimodular Locally Compact Groups
I have the following problem: "Let $G$ be a locally compact group, all of whose normal subgroups are contained in $Z(G)$. Prove that $G$ is unimodular."
My attempt at attacking the problem was to ...
4
votes
0answers
230 views
A condition on Fourier transforms that implies absolute continuity
Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
4
votes
0answers
134 views
Uniqueness of the random variable from its distribution
When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf ...
4
votes
0answers
452 views
Characteristic functions based proof problem.
I am trying to show that if $T$ be a closed bounded interval and $E$ a measurable subset of $T$. Let $\epsilon >0$, then there is a step function $h$ on $T$ and a measurable subset $F$ of $T$ for ...
4
votes
0answers
160 views
Existence of measures assigning positive values to all open sets
Let $K$ be a compact Hausdorff space. Does there exist a finite Borel measure on $K$, assigning positive values to all non-empty open sets of $K$?
4
votes
0answers
201 views
Fourier dimension of sets of positive Lebesgue measure
Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
4
votes
0answers
244 views
Proving the measure of an increasing sequence of measurable sets is the limit of the measures
Show that if $A_1\subseteq A_2\subseteq A_3\cdots$ is an increasing sequence of measurable sets(so $A_j\subseteq A_{j+1}$ for every positive integer $j$),then we have
...
4
votes
0answers
496 views
Dunford-Pettis Theorem
The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that:
A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact.
Now ...
4
votes
0answers
191 views
Why is the Radon-Nikodym derivative needed for products of complex measures?
Given $\mu, \nu$ complex Borel measures on $\mathbb{R}^n$, then product measure $\mu \times \nu$ on $\mathbb{R}^n \times \mathbb{R}^n$ is defined by
$$d(\mu \times \nu)(x, y) = \frac{d\mu}{d|\mu|} ...
4
votes
0answers
98 views
Convergence of integrals of Radon measures
Let $X$ be a locally compact Hausdorff space and let $\mu_n$ be a sequence of bounded variation Radon measures on $X$ such that $\int_X g \;d\mu_n \rightarrow \int_X g \;d\mu$ for each $g \in C_0 (X)$ ...
