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Questions tagged [measure-theory]

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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21 views

measure and countable set

how can we prove that if $\mu$ is a probability measure on $(\mathbb{R}^d,B(\mathbb{R}^d))$ then the set $E:=\left\{x \in \mathbb{R}^d,\mu(\left\{x \right\}) \neq 0\right\}$ is countable. Thank you.
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0answers
16 views

Function is Baire-1 if and only if these sets are $F_\sigma$

I AM LOOKING FOR A HINT, NOT A FULL SOLUTION, TO THE FOLLOWING PROBLEM: A function $f: [a,b] \to \mathbb{R}$ is called Baire-1 if it is the pointwise limit of a sequence of continuous functions. ...
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0answers
32 views

Why can we apply Fubini's theorem here?

Let $f : \mathbb{R^n} \rightarrow \mathbb R$ a measurable function define the function $t \mapsto \lambda_f(t) = \mathcal L\{ x : \vert f(x) \vert > t\}$ where $\mathcal L$ is the Lebesgue measure ...
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0answers
15 views

Explanation for Oxtoby's proof: a nonempty topological space $X$ is Baire iff player (I) has no winning strategy in the Choquet game

A nonempty topological space $X$ is a Baire space iff player I has no winning strategy in the Choquet game $G_X$. Oxtoby's proof I have several questions about this proof. $(\Leftarrow)$ How can ...
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1answer
22 views

Showing a function is in $ L^\infty$

Let $(X,\mu)$ a finite measure space and $g\in L^2_\mu$. Suppose there is a constant $K\ge 0$ s.t. for any measurable set $B$ , $|\left<g,\chi_B\right>|\le K \cdot \mu(B)$, how can I show that ...
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0answers
29 views

$f$ is measurable if and only if $E_i$ is measurable

Let $f:X\to\mathbb R$ be a simple function in the standard representation, $\displaystyle f=\sum_{i=1}^n\alpha_i\chi_{E_i}, $ where $\alpha_i\neq\alpha_j,E_i\cap E_j=\emptyset$ if $i\neq j$ and $\...
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0answers
34 views

left uniform continuous function that is not right uniform continuous

Consider the following problem 5 below. I am trying to construct the stated function f. I tried many functions one which sends the matrix of the form below to xy. I also tried to send the matrices of ...
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0answers
27 views

What is the base measure in measure theory?

I see the term "base measure" used frequently about measures. I do not completely get what that exactly means: Some examples are: Let $\cal F$ be the space of all probability density functions ...
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0answers
14 views

Can a stochastic process be neither adapted to filtration nor previsible?

The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'. If a process is adapted, it essentially means that the evolution of the ...
3
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2answers
37 views

Let $\mu(X)=1$, $0 \leq f \leq k$, and $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$.

Let $\mu(X)=1$ for $\mu$ a positive measure. Let $0 \leq f \leq k$ for some $k\in\mathbb{R}$ and let $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$. My attempt: I tried to expand ...
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0answers
17 views

Show that every measurable function $f$ is the limit of a sequence of continuous functions converging in measure.

This is a challenging question from my Analysis class: Given a measurable space and a finite regular measure defined on the $\sigma$-algebra, show that every measurable function $f$ is the limit of a ...
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0answers
40 views

about unmeasurable set on R

Suppose $\{E_n\}$ is a sequence of sets in $\mathbb{R}$, and $E_n$ are pairwise disjoint sets satisfying $$ m^*(\cup_{n=1}^{\infty}E_n)<\sum_{n=1}^{\infty}m^*E_n. $$ I want to prove that there ...
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0answers
9 views

Wave cone of the curl operator

How can one compute the wave cone $\Lambda_{\mathcal A}$, defined as \begin{equation*} \Lambda_{\mathcal A}:=\bigcup_{|\xi|=1} \ker \mathbb A^k(\xi) \qquad\textrm{with}\qquad \mathbb A^k(\xi)= (2\pi ...
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1answer
24 views

Continuity of Lebesgue integral of integrable function

I'm tackling the following question: My approach to (a) was: Let $x \in (0, \infty)$ and consider any sequence $x_j \to x$. We are asked to prove that $v(x_j) \to v(x)$ i.e. that $$\int_\mathbb R \...
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1answer
25 views

Compactness of the set of finite Borel measures

Suppose $X$ is a compact subset of $\mathbb{R}^n$ for some $n \in \mathbb N$. Let $\mathcal M(X)$ denote the space of all finite Borel measures on $X$. Is $\mathcal M(X)$ compact under some commonly ...
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1answer
36 views

Showing Lebesgue Integral inequalities

Let $f,g: [0,1] \longrightarrow (0,\infty)$ be measurable and $\beta >0$. Assume that $$\int_{0}^{1}g(x)dx = 1.$$ Show that $$1\leq \Bigg(\int_{0}^{1}f(x)^{-\beta}g(x)dx\Bigg)\Bigg( \int_{0}^{1}f(x)...
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0answers
32 views

Showing differentiability [on hold]

Let $\phi: \mathbb{R} \longrightarrow \mathbb{R}$ be a bounded differentiable function such that $\phi^{'}$ is bounded as well. Let $f \in L^1(\mathbb{R})$. Define $F:\mathbb{R} \longrightarrow \...
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1answer
22 views

For non $\sigma$-finite $d\mu$, integrable $f$, and $A_f = \{ f > 0 \}$, show that $\mathcal{V}(E) = \int_E \chi_{A_f}(x) \, d\mu$ is $\sigma$-finite.

Let $f \ge 0$ be integrable for a measure $d\mu$ e.g. $\int f \, d\mu < \infty$. Here $d\mu$ will not in general be $\sigma$-finite. Let $A_f = \{ x : f(x) > 0 \}$. Show that the set ...
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1answer
10 views

Distributive function of the sum of two measurable functions.

Let $(X, \mu)$ a measurable space and let $f,g:X \to \mathbb{C}$ complex measurable functions. We define the Distributive function of $f$ by $$D_f(\lambda) = \mu(\{x : |f(x)| > \lambda \}) $$ I'm ...
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0answers
48 views

Showing that a certain “norm-like” function fails to satisfy triangle inequality

Let $I$ denote the unit interval. Define $f: I \times I \to \mathbb{R}$ by $$f(x,y) = \begin{cases} 0 & x \in [0, \frac{1}{2}], \ y \in [0, \frac{1}{2}] \\ 0 & x \notin [0, \frac{1}{2}], \ ...
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2answers
62 views

Prove that $\mu((0,\infty))=0$.

Suppose $\mu$ is a measure on the Lebesgue measurable subsets of $\mathbb{R}$ and assume that there is a $K\geq 0$ such that for all $n\in\mathbb{N}$, we have $$\displaystyle\int_\mathbb{R} e^{nx}\ d\...
1
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2answers
32 views

Continuous functions, null sets and Lebesgue measurable sets

so im trying to prove that if i have a continuous function then f transforms null sets in null sets if and only if f transform Lebesgue measurable sets in Lebesgue measurable sets. Anyone has got some ...
0
votes
1answer
17 views

Find a function sarting with a Lebesgue-Stieltjes measure

Exercise. Find a non-decreasing and right-continuous function $F\colon\Bbb{R}\to\Bbb{R}$ whose Lebesgue-Stieltjes associated $\mu_F$ satisfies the following conditions at same time $\mu_F(\{0\...
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2answers
17 views

A collection that is an algebra and countable additivity

Let $X$ be a countable and infinite set and $$\mathcal{F}:=\{A\subset X;\#A<\infty\ \text{or}\ \#A^C<\infty\}.$$ (a) Show that $\mathcal{F}$ is an algebra. (b) Given $A\in\mathcal{F}$, ...
2
votes
1answer
34 views

Does the Lebesgue measure on the segment $y=x$ represent this distribution?

Set $\Omega=(-1,1)^2$. Consider the following measure on $\Omega$: $\mu(A)=m(A \cap L)$, where $L=\{ (x,x) \, | \, -1 < x < 1\}$ (the segment of the line $y=x$ in $\Omega$) , and $m$ is the ...
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2answers
32 views

coin/ sigma-algebra

You flip a coin two times. You consider two events: $$A=\{ " it \ lands\ heads \ up \ two \ times"\}$$ $$B=\{ " it \ lands\ tails\ up \ two \ times"\}$$ Which events do I have to add to get an ...
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0answers
63 views

about Lebesgue measure on R

E is a set such that $m(E)>0$, $E \subset (0,1)$ and there exist $c>0$ such that for some moving interval $I$, $$\lim_{mI\rightarrow0}\frac{m(E\cap I)}{m(I)}=c$$ Proof:mE=1 My attempt ,I have ...
2
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1answer
31 views

ODE with discontinuous vector field

Consider the ODE $$\partial_t \Phi(t,x) = \mathbf b(\Phi(t,x)), \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2$$ $$\Phi(0,x) = x, \quad x \in \mathbb R^2,$$ where $\mathbf b = (0,\chi_{\{x_1 \...
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0answers
27 views

A measure theoretic Lipschitz condition

Let $f$ be a measurable function satisfying following condition: for every $\epsilon$, we have \begin{equation*} \limsup_{\delta \to 0} \bigg\{ \frac{1}{\delta^N} \mathcal L^{2N} \Big( \Big\{ (x,y) \...
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0answers
29 views

Convergence in probability of running maximum

Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
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0answers
9 views

Finding a set that satisfies this property

i am looking for a set that satisfies this property but im lost , can you help me out ? $\underline{c}(E) < m^\star(E) < \overline{c}(E)$ where this $c$ is referring to the jordan content. I ...
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0answers
30 views

Random variable defined on the Lebesgue probability space

There is a random variable X defined on the Lebesgue probability space whose cumulative distribution function is F. We can find X(w) knowing that: $X(ω)=\inf\{x∈R:F(x)>ω\}$. 1) how do we prove ...
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1answer
32 views

Borel measurability of a set

Hey guys I have a question on the Borel measurability of this set $\{{(x,y):x∈E,0<y<f(x)}\}$ when $f$ is a continuous function defined in an open set. Can anyone help me out? I think the set is ...
1
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1answer
18 views

Bounded, tight sets of measures are compact?

In Prokhorov's 1956 paper "Convergence of Random Processes..." it states the following. Where $\mathcal{R}$ is a complete, separable, metric space. Additionally, it says that any weakly convergent ...
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25 views
+50

Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative? More precisely, I'd like to see an example of a function $$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$ ...
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0answers
30 views

A doubt on masa

Let $g$ in $L^{2}[0,1]$, let $f_{n}$ are in $L^{\infty}[0,1]$ such that $f_{n}{\rightarrow}^{\|\cdot\|_{2}} g$, from $f_{n}$ can we construct $g_{n}$ such that $g_{n}\rightarrow g$ in $\|\|_{2}$ norm ...
1
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1answer
28 views

Why is the first return map measure preserving?

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ Definitions and Context Let $(X, \mc X, \mu, T)$ be an invertible measure preserving system and let $A$ be a measurable subset of $X$ with ...
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1answer
17 views

Doubt about a conterexample about the Lebesgue Dominated Convergence Theorem

just a quick question to check if this is right. So i have been trying to find a counterexample when we cant swap the limit with the integral when the fuction isnt dominated and i was wondering if ...
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1answer
20 views

Question about Radon measures in $\mathbb{R}^n$

Consider the following: Let $f$ be a $L^1(\mathbb{R}^n)$ function and define the Borel measure $d\mu=fdx$, where $dx$ is the Lebesgue measure. Now, since $f$ is in $L^1(\mathbb{R}^n)$, we know that $...
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1answer
47 views
1
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1answer
18 views

Haar Measures and Embeddings of $\nu$-adic integers

Let $\nu\geq4$ be any composite integer, and let $d\in\left\{ 2,\ldots,\nu-1\right\}$ be any non-trivial divisor of $\nu$. Since $d\mid\nu$, note that any sequence $\left\{ \mathfrak{y}_{n}\right\} _{...
6
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1answer
108 views
+150

Understanding Optimal Transport in One Dimension.

I'm trying to understand these lecture notes. https://sites.ualberta.ca/~mathirl/IUSEP/IUSEP_2018/lecture_notes/Pass1.pdf I understand the formulation of the Monge Problem. However, I'm having trouble ...
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1answer
20 views

How to change the order of the limit and the expectation?

Does anyone know how to prove $\lim E[X(n)]=E[\lim X(n)]$??? Here I need to prove $\lim E[X(n)]\le E[\lim X(n)]$ and $\lim E[X(n)]\ge E[\lim X(n)]$. Based on "Fatou Lemma", I can get that $E[\...
2
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0answers
19 views

Gluing together 2-dimensional martingale measures to create n-dimensional martingale measure, Strassen's Theorem

Strassen's theorem states that a necessary and sufficient condition for existence of a discrete-time martingale with a finite number $n$ of given marginals $\mu_1,\ldots,\mu_n$ is that the marginals ...
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0answers
10 views

Calculation of area of a disk directly from Lebesgue measure properties.

I'm currently stuck upon an apparently "easy" problem: calculate the area of a real disk using the theory of the Lebesgue measure in $\mathbb{R}^n$. I say "easy" because, using standard integration ...
3
votes
1answer
52 views

Let $m^\ast(E)$ < ∞.If for every interval (a, b) we have that $b-a$=$m^\ast((a,b)∩E) + m^\ast((a,b) ∩ E^c)$ then $E$ is lebesgue measurable

Since we are considering only few one type if sets (i.e open intervals) , i don't know how to prove $m^\ast(A)$=$m^\ast((A∩E) + m^\ast(A∩ E^c)$ using only information that $b-a$=$m^\ast((a,b)∩E) + m^\...
0
votes
1answer
46 views

Is there an invariant measure absolutely continuous wrt to the lebesgue measure for the map f

Let $f:[0,1]\rightarrow[0,1]$ where $f(x)=x/2$ $(1-x)$, and let $\lambda$ be the lebesgue measure on [0,1]. Is there a probability measure $\mu$ that is invariant and absolutely continuous wrt to the ...
3
votes
1answer
68 views

Examples of Lebesgue-integrable, but not Riemann-integrable functions

The standard example of this is the characteristic function of the rationals. However this is somewhat pathological as this function is zero almost everywhere. What are other examples that differ from ...
2
votes
1answer
42 views

$\nu<<\mu$ where $\nu$ is arbitary signed measure and $\mu$ is $\sigma$- finite measure then $\exists f$ such that $d\nu =fd\mu$

If $\nu<<\mu$ where $\nu$ is arbitary signed measure and $\mu$ is $\sigma$- finite measure then $\exists f$ which is extended $\mu $ integrable $f:X\to [-\infty,\infty]$ such that $d\nu =fd\mu$ ...
0
votes
2answers
23 views

Is the diagonal in the product $\sigma$-algebra $\mathcal{P}(\mathbb{R})\otimes\mathcal{P}(\mathbb{R})$?

Is the diagonal $\Delta_\mathbb{R}:=\{(x,x):x\in\mathbb{R}\}$ in the product $\sigma$-algebra $\mathcal{P}(\mathbb{R})\otimes\mathcal{P}(\mathbb{R})$, where $\mathcal{P}(\mathbb{R})$ is the power set ...