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

learn more… | top users | synonyms (1)

2
votes
1answer
18 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
0
votes
1answer
15 views

Infimum of an outer measure

I'm stuck in this problem. Let I a set of index and $m_i$ outer measure for all $i\in I$. Show that $\inf\limits_{i\in I}m_i$ is an outer measure. Specifically I don't know how to show the ...
0
votes
0answers
24 views

Lebesgue integration; convergence in measure [on hold]

Suppose ${f_n}$ is a sequence of measurable real functions on $[0,1]$ and $\int f_n^2 \leq 1 \: \forall n$. Further, suppose $f_n \to 0$ in measure. Show $\int f_n \to 0$.
0
votes
1answer
40 views

Find $\liminf X_n$ where $X_n=1_{[n,n+1]}$?

My attempt: Suppose $\omega=n_0$. Then choose $N\geq n_0+1$.Threfore, $X_N(\omega)=0$. Therefore, $\inf_{k\geq N}X_k(\omega)=0$. Does it suffice to prove that $\liminf\limits_{n \rightarrow \infty} ...
0
votes
2answers
61 views

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials?

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials? And why? I know it is the uniform limit on a set take out some finite measurable set but not sure if I can say more. Thanks.
0
votes
1answer
16 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
1
vote
0answers
24 views

A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu $ and $\lambda $ are measures in $M$ st $\mu \ge $ $\lambda $ then show that $m$ defined as $\mu= \lambda + m $ is a measure 2) Prove that if ...
1
vote
1answer
18 views

Prokhorov-like convergence

Let $(X,d)$ be a metric space, and for any $A\subseteq X$ define $$ A^\delta:=\{y\in X:\exists x\in A \text{ such that }d(x,y)\leq \delta\}. $$ Under which conditions on $(X,d)$, $A \subseteq X$ and ...
0
votes
1answer
24 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
1
vote
1answer
27 views

How to show $\displaystyle s=\sum_{i=1}^n \alpha_i \chi_{A_i}$ measurable implies $A_i$ measurable for all $i=1, \ldots, n$?

Let $X$ be a measurable set and $s:X\longrightarrow [0, \infty)$ be a simple function. It is easy to see $$s=\sum_{i=1}^n \alpha_i \chi_{A_i},$$ where $\{\alpha_1, \ldots, \alpha_n\}$ is the set of ...
0
votes
2answers
76 views

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
3
votes
1answer
72 views

Measures $\mu$ such that $\mu(a+A)\leq c\ \mu(A)$

Let $\mu$ be a positive measure on $\mathbb{R}$ such that $\mu[a,b]<+\infty$, for all $a,b\in\mathbb{R}$ and $\mu(\mathbb{R})=+\infty$. The set $a+A$ denotes the translation set of $A$ by a, i.e. ...
0
votes
1answer
38 views

Inclusion for pairwise disjoint sets and $\limsup A_n = \emptyset$

Spin-off from here. 1 Please give an example of how the following does not hold for a collection that is not pairwise disjoint. $$ \bigcup_{k \ge n+1} A_k = A\setminus (A_1 \cup\cdots \cup A_n) $$ ...
1
vote
0answers
51 views

$X_n=n^21_{(0,1/n)}$. What is $\limsup\limits_{n \rightarrow \infty} X_n$

$X_n=n^21_{(0,1/n)}$. What is $\limsup\limits_{n \rightarrow \infty} X_n$? What about its $\liminf\limits_{n \rightarrow \infty} X_n$? My attempt: For each $n$ on $\{0, [1/n,1]\}$, we have ...
1
vote
0answers
38 views

Prove that E$(\liminf\limits_{n\rightarrow \infty} X_n )\leq \liminf\limits_{n\rightarrow \infty}E(X_n)$

I want to prove that if $X_n\geq 0$, then E$(\liminf\limits_{n\rightarrow\infty} X_n )\leq \liminf\limits_{n\rightarrow\infty}E(X_n)$. My attempt: $E(\liminf\limits_{n \rightarrow \infty} ...
0
votes
0answers
7 views

Extension of premeasure to outer measure

I'm stucked in this problem. Show that all outer measure $\mu_*$ can be expresed of the form $$\mu_*(A)=\inf_{A\subset\cup C_i}\sum \tau(C_i)$$ where $\tau$ is a premeasure of a colection of sets ...
2
votes
1answer
28 views

Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?

Let $C$ is a compact subset of $\mathbb R,$ $V\subset \mathbb R,$ and $0<m(V)<\infty,$ where $m$ is a Lebsgue measure on $\mathbb R.$ My Question is: Can we expect to find $k\in ...
4
votes
0answers
46 views

Jordan decomposition of linear functionals

Let $X$ be a locally compact Hausdorff space. Also, let $C_0(X,\mathbb R)$ denote the vector space of such continuous functions $f:X\to\mathbb R$ that the set $\{x\in X\,|\,|f(x)|\geq\varepsilon\}$ is ...
1
vote
1answer
18 views

convergence in measure of min $(f_n,g)$

I was reading a proof of a convergence in measure variant of fatou's lemma earlier and there was a seemingly easy part of it I just could not verify. Assume $(f_n)_{n \in \mathbb N}$ is a sequence of ...
1
vote
1answer
23 views

Almost Trivial $\sigma-$fields

I am trying to understand the proof of the following Lemma form the book A probability path by sidney Resnick. Lemma: Let $\mathcal{G}$ be an almost trivial $\sigma-\text{field}$ and let $X$ be a ...
0
votes
4answers
92 views

Are null sets necessarily closed?

Hi everyone: Is a null set of $\mathbb{R}^n$, $(n>0)$, necessarily closed? Give a counter example. Thanks for your reply.
1
vote
1answer
30 views

Complex measures vs. Positive Measures

In his real and complex analysis, Rudin writes that the right hand side of the expression $\mu(E) = \Sigma \mu(E_i)$ must necessarily converge for any countable partition $\{E_i\}$ of a measurable E, ...
1
vote
2answers
41 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
0
votes
1answer
28 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \ mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
9
votes
1answer
120 views

Measure which does not grow faster than Lebesgue

Is there an example of a measure $\mu$ on $\mathbb{R}$ which is not absolutely continuous with respect to Lebesgue measure such that $\mu[\mathbb{R}]=+\infty$ but $$\limsup_{a\to ...
1
vote
1answer
23 views

Existence of a sequence of continuous functions converging pointwise to a characteristic function.

I'm reading Rudin's Real and Complex Analysis and in section 5.11 he makes the next assertion: Put $g_n(t)=1$ if $D_n(t)\geq 0$, $g(t)=-1$ if $D_n(t)<0$. There exist $f_j\in C(T)$ such that ...
7
votes
1answer
32 views

If $w$ is in weak $A_{\infty}(d\mu)$ where $d\mu$ is a doubling measure, then is $w\,d\mu$ doubling?

Let $\mu$ be a positive Borel measure on $\mathbb{R}^n$ and let it be doubling i.e. there exists a a constant $C>1$ such that $\mu(B(x_0, 2r)) \leq C \mu(B(x_0,r))$ for all balls $B(x_0,r)$. Let ...
0
votes
1answer
71 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
0
votes
0answers
31 views

Limit of measurable function is measurable

This question has been asked already here but I didn't get a satisfactory solution and didn't want to bring up an old question. Here is the question : Let $\{f_n\}$ be a sequence of measurable ...
2
votes
1answer
31 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
1
vote
2answers
37 views

The applicability of the Dominated Convergence theorem on the real line

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it ...
0
votes
1answer
22 views

Convergence in Probability implies Weak Convergence Proof Question

I'm trying to follow a proof for showing $\displaystyle \lim_{n\rightarrow \infty} P[|X_n-X|>\epsilon] = 0 \Rightarrow X_n \rightarrow_p X$ The first step of the proof says: $P[X \leq x-\epsilon] ...
1
vote
1answer
35 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
0
votes
0answers
23 views

Measure Theory - working with unusual measures and set functions

Let $m$ define the Lebesgue measure. Let $\mu$ define the measure $\mu(A)=m(A\cap(0,1))$ for a Borel set $A$. Let $K=\bigcap \{A:A$ is closed, $\mu(A)=1\}$, $D=\bigcap \{G:G$ is open, ...
0
votes
1answer
31 views

Step function converging to $f$

Let $(X,S,\mu)$ be a $\sigma$-finite measure space, and $f:X\to\mathbb{R}$ a measurable function with $f(x)\geq 0$ for all $x\in X$. Show that there exists a sequence of step functions, $\{ \phi_n\}$, ...
4
votes
0answers
40 views

A formula similar to $\int_a^bf(x)dx=\mu\left[a,b \right]$ for $f^p$.

Let $\mu$ be an absolutely continuous measure with respect to the Lebesgue measure on $\mathbb{R}$ , and $f:\mathbb{R}\to \mathbb{R^+}$ its Radon-Nikodym derivative . We can write $\int_a^bf(x)dx$ in ...
0
votes
1answer
34 views

Given a pairwise disjoint collection, $\limsup A_n = \emptyset$?!

Let $(A_n)_{n=1}^{\infty}$ be a pairwise disjoint collection. $\limsup A_n = \emptyset$?! This is in relation to @StephenMontgomery-Smith's hint here. Trying prove $\sigma$-additivity from ...
0
votes
1answer
39 views

Prove that a Modified Cantor Distribution is Atomic.

Consider a measurable space $\{\mathcal{I},\mathcal{B}\}$, where $\mathcal{I} = [0,1]$ and $\mathcal{B}$ are the Borel sets on $\mathcal{I}$. And also, denote $\mathcal{C}$ as the cantor set on ...
2
votes
1answer
64 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
0
votes
0answers
30 views

Inequality of integral of a function over cubes in $\mathbb R^n$ [closed]

Let $n>1$. For any $M>1$, show that there exists $C_M>1$ such that $$\int\limits_{MQ}\frac{1}{|x|}\, dx \leq C_M\int\limits_Q \frac{1}{|x|}\,dx$$ for any cube $Q\subset \mathbb{R}^n$ where ...
2
votes
1answer
43 views

Nonmeasurable functions and the axiom of choice

Hi everyone: We know that to construct a nonmeasurable function one must to use the axiom of choice. Can we conclude that to avoid all nonmeasurable functions it suffices to reject the axiom of ...
2
votes
2answers
35 views

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ [duplicate]

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ Not sure how to go about this problem. I tried Fubini. But that ...
0
votes
1answer
29 views

Prove if $E$ is a Lebesgue measurable set, there exists a continuous function $f$ differing from $\chi_{E}$ on a set of measure $< \epsilon$?

I am reviewing my analysis notes, and I don't really understand the proof given by my professor. He first proved if $E$ is a Lebesgue measurable set and $\epsilon > 0$, then there is an open set ...
0
votes
0answers
13 views

For $k ∈ \mathbb{Z}$ and for $x ∈ [k/n, (k + 1)/n)$ set $g_n(x) = n\int_{k/n}^{\frac{k + 1}{n}}f(x)dx$.

Let $f ∈ L_1(\mathbb{R}).$ For $n ∈ \mathbb{N}$ define the function $g_n :\mathbb{R}→\mathbb{R}$ as follows. For $k ∈ \mathbb{Z}$ and for $x ∈ [k/n, (k + 1)/n)$ set $g_n(x) = n\int_{k/n}^{\frac{k + ...
1
vote
1answer
30 views

Simulation of a random vector

I have a question which is probably well known but I do not find any written reference. Let us consider a probability measure $\mu$ on $\mathbb{R}^2$. I would like to know if one can find a random ...
6
votes
1answer
54 views

Haar Measure for Algebraic Number Theory: What Should I Know?

I recently taught myself some algebraic number theory and am preparing to take a course in class field theory this fall. I understand the notion of a Haar measure on a locally compact topological ...
1
vote
1answer
19 views

How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
0
votes
1answer
30 views

Metrization of the weak topology on the set of radon measures

Let $\mathcal{M}$ denote the set of Radon measures on $\mathbb{R}$. We endow $\mathcal{M}$ with the the weakest topology such that $\mu \to \int f \, \mathrm{d} \mu$ is continuous for all $f \in ...
1
vote
0answers
10 views

How to define a convex hull on probability measure set? [closed]

In the finite dimensional space $R^n$, the convex hull of a set $C$ can be represented by a convex combination of the point in $C$. Especially, the Caratheodory's theorem the number of point is at ...
0
votes
1answer
19 views

Return Lemma MC

If a Markov chain is $\phi$-irreducible and has stationary distribution $\pi$, then $\phi\ll \pi$, Proof: We use the irreducibility of the chain to write the state space $E = \bigcup_{n,m \in ...