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

learn more… | top users | synonyms (1)

3
votes
0answers
22 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{r>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to show ...
2
votes
2answers
25 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
0
votes
0answers
10 views

Definition of an Algebra - Measure Theory

So an algebra of a fixed set $X$ is a collection of subsets of $X$ such that it is closed under complementation and unions of sets. So is the difference between an algebra and a $\sigma$-algebra ...
0
votes
0answers
13 views

Outer Measures - Measure Theory

In the definition of an outer measure, they state the sub-additivity condition as $\mu_{*}(\bigcup A_{n}) \leq \sum\mu_{*}(A_{n})$ for any sequence of sets $A_{n} \subset X$ My question is does ...
2
votes
1answer
29 views

Application Birkhoff ergodic theorem

Let $(X,\mathcal{B},m,T)$ be a probability preserving transformation. Let \begin{align*} I:&=\{f\in L^1: f=f\circ T\}\\ B:&=\{g-g\circ T: g\in L^1\} \end{align*} I have to show that $$ ...
0
votes
0answers
7 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
1
vote
1answer
27 views

Probability of tail event using Kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
2
votes
2answers
55 views

Lebesgue measure of graph of $\sin{\frac{1}{x}}$ on $[0,1]$

I am working on something and read that measure of graph of a continuous function on compact sets is zero. Now, I tried to do it for non continuous functions but the set of discontinuities have ...
1
vote
3answers
59 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
0
votes
0answers
5 views

Closed subgroup of a locally compact Hausdorff group whose Haar measure is non-zero.

Let $G$ be a locally compact Hausdorff group, $H$ its closed subgroup. To avoid pathologies, we assume the underlying topological space of $G$ has a countable base. Let $\mu$ be a Haar measure on $G$. ...
1
vote
0answers
12 views

Weil's definiton of image of Haar measure on homogeneous space $G/\Gamma$ where $\Gamma$ is discrete

Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. Let $\Gamma$ be a discrete subgroup of $G$, $G/\Gamma$ the ...
8
votes
2answers
110 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
1
vote
0answers
24 views

Defining Lebesgue measure on a subspace of $\mathbb{R}^n$

Let $\bar{w}_1,.., \bar{w}_k$ be linearly independent vectors in $\mathbb{R}^n$. Let $W$ be the subspace spanned by these $\bar{w}_i$'s. I know how the Lebesgue measure is defined on $\mathbb{R}^n$. ...
0
votes
1answer
39 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...
6
votes
1answer
1k 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 ...
0
votes
0answers
11 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
0
votes
1answer
20 views

Example of strict inequality in special case of fatou's lemma.

Give an example of sequence of events $\{A_n\}$ such that the following inequalities are strict $P(\lim\inf A_n) \le \lim\inf P(A_n) \le \lim\sup P(A_n) \le P(\lim\sup A_n)$. Thanks
0
votes
0answers
15 views

Weak Compactness Therem for $L^p$

I have a problem to understand a point in the Weak Compactness Theorem from the book of Evans/Gariepy. So we have $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ a Radon measure with $\mu \ll ...
0
votes
1answer
10 views

Borel Measurable Function 3

So the definition of a measurable function is as follows: Let $f: (X, \mathcal{A}) \rightarrow\overline{\mathbb{R}}$ be a function on the measurable space $(X, \mathcal{A})$. Then $f$ is said to be ...
0
votes
1answer
21 views

Open Sets in the Extended Real Line

So I know that the extended real line is given by $\mathbb{R} \bigcup$ {$-\infty, \infty$}. So these are the facts that I know: 1) Firstly, every interval in $\mathbb{R}$ is a Borel Set (I seem to ...
3
votes
0answers
26 views

Borel $\sigma$-algebra

Since the Borel $\sigma$-algebra is generated by the family of open sets, does that mean that every Borel set is essentially some countable union/intersection of open sets or a complement of open ...
2
votes
1answer
52 views

Prove Y = X given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$

Prove Y = X, given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$ Attempt: Suppose $Y = E[X|\mathscr{G}] $. Then $E[X|\mathscr{G}] $ is $\mathscr{G}$-measureable. For every A $\in \mathscr{G}$: ...
0
votes
2answers
19 views

Borel Sigma Algebra generated by Open Intervals

So I know that the Borel $\sigma$-algebra of $\mathbb{R}$ is the $\sigma$-algebra generated by open sets. I have been able to prove that this Borel $\sigma$-algebra is also generated by the family of ...
1
vote
0answers
28 views

Marking Integers Using a Wheel

Suppose I had a wheel of diameter one meter and I was charged with marking every meter along an infinite stretch of a beach. The strategy is to insert pegs into the wheel so that every point that is a ...
0
votes
1answer
32 views
1
vote
1answer
30 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
0
votes
1answer
27 views

Prove a sequence of integrals converges to 0

Let $E$ be a set of finite Lebesgue measure in ${\bf R}$ and $\{a_n\}_{n \in {\bf N}}$ be a sequence of real numbers. Show that $\int_E \cos(nx + a_n) dx $ goes to 0 as $n \to \infty$. I tried ...
1
vote
0answers
60 views

Check proof that if $\|f\|_{L^{\infty}{(\Omega)}}=0$ then $f=0 $ a.e on $\Omega$

Prove that for $p=\infty$ : $\|f\|_{L^{\infty}{(\Omega)}}=0 \implies f=0 $ a.e on $\Omega$ Proof $\|f\|_{L^{\infty}{(\Omega)}}=0 \implies \mathrm{ess}\sup |f| =0 \implies \inf\{a\in R| ...
4
votes
1answer
131 views

Analogue of Lebesgue differentiation theorem in Orlicz spaces

It is well known that $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all $x\in\mathbb{R}^{n}$. Here ...
1
vote
1answer
23 views

Integrability of dirichlet function in $\mathbb{R}^3$

Let $d: [0,1] \rightarrow \mathbb{R}$ be the Dirichlet function as follows: $$d(x) = \begin{cases} 1, & x \in \mathbb{Q} \\ 0, & x \in \mathbb{R} \backslash \mathbb{Q} ...
3
votes
1answer
30 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
1
vote
0answers
16 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
1
vote
0answers
17 views

Distribution of a r.v. with the same mean and variance is abs. cont. with resp. to the normal distr.

I have a question concerning the Kullback-Leibler divergence or relative entropy. In a book I found the following definition of the KL-divergence: Let $(\Omega, \mathcal F)$ be a measurable space. ...
0
votes
1answer
14 views

Finite measure space & sigma-finite measure space

A measure space $(X, \Sigma, \mu)$ is finite if $\mu(X)<\infty$. It is equivalent to saying that $(X, \Sigma, \mu)$ is finite if $\mu(E)<\infty$ for all $E \in \Sigma$ A measure space $(X, ...
0
votes
1answer
24 views

Is the support of the Gaussian finite or infinite?

Considering that as $x \to \pm \infty$ ; $e^{-\frac{x^2}{2}} \to 0$, is the support finite or infinite? A simple enough question, but enough to make me scratch my head. I feel that it's almost a ...
1
vote
2answers
53 views

Is a subspace of functions that essentially depend only on one variable closed?

Let $S$ be the subspace $$\left\{f\in L^p( I^2)|\exists g\in L^p( I), f(x,y)=g(x), \mbox{a.e. } (x,y)\in I^2\right\}.$$ Is $S$ closed under the $L^p$ norm? I think the first step would be to ...
1
vote
1answer
37 views

Finite meaure space with $f \in L^p$ [duplicate]

Given a finite measure space $(X,\Sigma,\mu)$, for $1<p<\infty$, if $f \in L^p(X)$, then $f \in L^1(X)$. Can anyone show me how to start the proof? Thanks.
0
votes
1answer
19 views

Can ratios similar to those related to the surface area of a circle and sphere be applied to determine properties of a 3-sphere?

Applying the strategy of describing the surface area of a circle as a product of the ratio for the surface area of a triangle, reveals a consistency that also applies to the surface area of a cone. ...
4
votes
1answer
289 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 ( $\sigma\ll\mu$)?
29
votes
2answers
517 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
2
votes
1answer
17 views

Demonstration that pure point set is countable if the measure is finite for every compact

I read on Reed and Simons' this statement. Let P be the pure point set of a positive Borel measure $\mu$ on $\mathbb{R}$, that is $P = \{ x\, |\, \mu(\{x\}) > 0 \}$. Then this set is countable if ...
2
votes
2answers
117 views

How does Wikipedia's definition of the Lebesgue integral relate to more common definitions?

Wikipedia presents a definition of the Lebesgue integral (of a nonnegative function $f$) that I hadn't encountered before: Let $f^*(t)=\mu \left (\{x\mid f(x)>t\} \right )$. The Lebesgue ...
5
votes
2answers
258 views

Lebesgue non-measurable function

Can we give an example of Lebesgue non-measurable function, for which set $\{x: f(x)=C\}~\forall C\in\mathbb{R}$ is measurable? Thanks.
0
votes
0answers
33 views

Measurability of the set of all differentiable points

Given a function $f:(a,b)\to\mathbb{R}$, is the set of all points where $f$ is differentiable a measurable set? Please provide a short proof or a counterexample. And furthermore, in case it's false, ...
0
votes
1answer
21 views

Borel Sigma Algebra- Measure Theory

I have tried looking at various sources and still cant understand the following: Consider $X= \mathbb{R}$, then the $\sigma$-algebra generated by the family of closed intervals $[a,b]$ is the same as ...
0
votes
2answers
32 views

Borel $\sigma$ -algebra

So I have a proposition which states the following: Each of the following families of sets generate the Borel $\sigma$-algebra: 1)The family of all open intervals $(a,b)$, $a,b, \in \mathbb{R}$. 2) ...
1
vote
0answers
9 views

Measure of convolution

Let $M$ be the Banach space of all complex Borel measures on $R$.The norm in $M$ is $\|\mu\|=|\mu|(R)$,associate to each Borel set $E\subset R$ the set $$ E_2=\{(x,y):x+y\in E\}\subset R^2 $$ if ...
0
votes
0answers
15 views

Reconstructing a measure from its (absolutely continuous) marginals

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation ...
4
votes
1answer
35 views

Is the following statement true on $L^0$ spaces?

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $X,Y\in L^0(\Omega;\mathbb{R})$ two random variables taking values in $\mathbb{R}$. Is it true that: $$\int_{A} f(X(\omega)) P(d\omega) = ...