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

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2
votes
2answers
49 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...
2
votes
0answers
35 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
1
vote
1answer
31 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
3
votes
1answer
75 views

Usual convex combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
3
votes
0answers
26 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
1
vote
0answers
13 views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
3
votes
0answers
61 views
+50

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
2
votes
1answer
48 views
+100

Extensions of universal measures

Let $(\Omega,\mathcal F)$ be a measurable space, and let $\mathcal P$ be the set of all probability measures no this space. Let $\mathcal F^p$ denote a completion of $\mathcal F$ w.r.t. $p\in P$ and ...
1
vote
1answer
26 views

Don't understand a $L^\infty$ bound argument involving measure of set

I'm trying to understand the proof of Proposition 2.2, part 2 of this paper. this is where I am stuck. For any $k > 0$, we have $$k^{\frac{2(N+1)}{N}}|\{|u|^m > k\}| \leq ...
2
votes
1answer
32 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
1
vote
0answers
27 views

Measurability and composition of functions

If $f\circ p=h$ and $p:S^m \rightarrow S$ the $i{\text{th}}$ projection ($S$ measurable space) and $f:S\rightarrow \mathbb{R}$ any function whatsoever ($\mathbb{R}$=reals) and $h$ measurable, is there ...
1
vote
1answer
22 views

Question on $x$-section of measurable rectangle in product measure space $X \times Y$

I'm reviewing my analysis notes. We have that $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces. We are considering the product measure space $(X \times Y, \Sigma(\lambda^{*}), ...
4
votes
2answers
53 views

Borel measure supported on $\mathbb{Q}$

Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?
3
votes
2answers
33 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
3
votes
0answers
39 views

Measurability of $\int f(x,\bullet) d\nu$ (product space)

I'm self-studying a set of notes on measure theory. For this chapter, the author said that he is not going to prove every theorem, and I have some trouble with this particular one: Theorem 5.15. Let ...
0
votes
2answers
46 views

Representation of linear functionals on a certain Banach space

Let $C^k([0,1])$ be the space of such complex-valued functions on $[0,1]$ that are continuously differentiable at least $k$ times ($k\in\mathbb N$). It is well known that $C^k([0,1])$ is a Banach ...
1
vote
1answer
36 views

Intersection of countable many sets of measure $1$

Consider a probability space $(X,\mathscr M,\mu)$ and a collection of measurable sets $\{A_n\}_{n\in\mathbb N}$ such that $\mu (A_n)=1$ for every $n$. Then I don't unterstand the following result: ...
2
votes
1answer
45 views

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

Problem: Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure. My Attempt: Let ...
2
votes
1answer
42 views

When does intersection of measure 0 implies interior-disjointness?

If there are two "nice" shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their ...
3
votes
1answer
66 views

$f$ is in $L^p$ iff sum is finite

Let $p\in [1,\infty)$.Prove that $f\in L^p(\mu)$ if and only if $\sum_{n=1}^\infty(2^n)^p\mu (\{x:|f(x)|\gt2^n\})\lt \infty.$ My idea, I assume measure is finite, I wrote ...
5
votes
1answer
38 views

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f_n||_1 = 1$. Set $F(x) = \sup_{n \in \mathbb{N}}f_n(x)$. Prove that $\int_\mathbb{R}F(x)dx ...
3
votes
1answer
18 views

$L^p$ integral on every measurable subset of $\Bbb R$

Suppose $f:\Bbb R \to \Bbb R$ is in $L^p$ for some $p>1$ and also in $L^1$. Prove there exist constants $c>0$ an $\alpha \in (0,1)$ such that $\int_A|f(x)|dx\le cm(A)^{\alpha}$, for every ...
3
votes
2answers
56 views

finite additivity&countable additivity

Let $\tau$ be a semialgebra of subsets of $\Omega$ and let P: $\tau\rightarrow [0,1]$, with $P(\Omega)=1$, and it satisfies finite additivity: $P\big(\bigcup_{i=1}^{n}D_i\big)=\sum_{i=1}^{n}P(D_i)$ ...
0
votes
2answers
26 views

Borel $\sigma$-algebra of subsets of [0,1]

Let sample space be [0,1]. let $\tau$ be the collections of all the open intervals on [0,1]. Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing $\tau$. Is the statement any ...
1
vote
1answer
29 views

Caratheodory extension theorem: which is the “unique extension”

According to Wikipedia, the constructed measure on $\sigma(R)$ is a unique extension. However, in most situations, the $\sigma$-algebra of Caratheodory-measurable sets $M$ is larger than $\sigma(R)$. ...
1
vote
1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
1
vote
1answer
40 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
3
votes
1answer
28 views

countably additive function P

This problem comes from exercise 1.3.5(b) of 'A First Look at Rigorous Probability Theory'. It asks to give an example of a countably additive function $P$, defined on all subsets of $[0,1]$, which ...
3
votes
0answers
45 views

Image of Cantor set under Cantor-Lebesgue function

Let $m^{\ast}$ be the Lebesgue outer measure and $m$ the Lebesgue measure. Let $\phi$ be the Cantor Lebesgue function and let $\psi(x) := x + \phi(x)$. Let $C$ be the standard Cantor set, why does ...
4
votes
2answers
102 views

Evaluating $\lim_{n \rightarrow \infty} \int^{n}_{0} (1+\frac{x}{n})^{-n} \log(2+ \cos(\frac{x}{n})) \> dx$

The problem I am stuck on asks the reader to find the following limit: $$\lim_{n \rightarrow \infty} \int^{n}_{0} (1+\frac{x}{n})^{-n} \log(2+ \cos(\frac{x}{n})) \> dx.$$ The section I am working ...
0
votes
1answer
20 views

Finite union of measurable rectangles can be written as union of pairwise disjoint measurable rectangles?

Suppose we have two measure spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$. $R \subseteq X \times Y$ is called a measurable rectangle if $R = A \times B$ with $A \in \Sigma$ and $B \in \tau$. I have ...
3
votes
1answer
28 views

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I ...
1
vote
1answer
36 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ...
2
votes
0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
2
votes
1answer
45 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
4
votes
1answer
56 views

Intersection of sets of positive measure

If $E$ and $F$ are sets of positive Lebesgue measure on $\mathbb{R}$, prove that some translate of $F$ intersects $E$ in a set of positive measure. Since each are approximated from below by compact ...
4
votes
3answers
106 views

Find $\delta >0$ such that $\int_E |f| d\mu < \infty$ whenever $\mu(E)<\delta$

I am studying for a qualifying exam, and I am struggling with this problem since $f$ is not necessarily integrable. Let $(X,\Sigma, \mu)$ be a measure space and let $$\mathcal{L}(\mu) = \{ \text{ ...
2
votes
1answer
48 views

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that ...
1
vote
2answers
40 views

Jordan measure zero discontinuities a necessary condition for integrability

The following theorem is well known: Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ...
0
votes
0answers
38 views

find an adjoint operator for operator N [closed]

I read this in a paper how to find its adjoint?
0
votes
0answers
20 views

essential infimum

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, if $u\in L^{\infty}(\Omega)$ and we want define the infimum of $u$ we write $\mathrm{ess}\inf_{x\in\Omega} u(x)$ but if $u\in ...
0
votes
0answers
32 views

Singular measures

Suppose $\mu$ and $\nu$ are two signed measures and they are mutually singular. I heard 2 version on this definition there $\exists N $ s.t.$\mu(N)=0$ and (1) $\nu(N^c)=0$ (2)$|\nu|(N^c)=0$ which ...
0
votes
0answers
31 views

Measure of $\chi_\mathbb{Q}(x)$?

$\chi_\mathbb{Q}(x) = 1$ if $x \in \mathbb{Q}, 0$ otherwise. Well $\chi_\mathbb{Q}(x)$ is a measurable function if $\mathbb{Q}$ is a measurable set. $\mathbb{Q}$ is a measuable set under the Borel ...
1
vote
1answer
40 views

A problem in the proof of Jordan decomposition theorem

How to obtain the red rectangle in the picture? thanks in advance.
0
votes
1answer
37 views

Denseness and non-measureability

I made up a question for myself and tried to answer it. I'm not completely sure of my question and answer, since I lack a grounding in analysis (the tragedy of doing physics then mathematical ...
2
votes
1answer
61 views

Can we find uncountably many disjoint measurable subsets of $\mathbb{R}$ with strictly postive Lebesgue measure?

Can we find uncountably many disjoint measurable subsets of $\mathbb{R}$ with strictly positive Lebesgue measure?
2
votes
1answer
41 views

Unclear inequality in the proof of Birkhoff ergodic theorem.

I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ...
1
vote
1answer
36 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
1
vote
0answers
52 views

Why $\int f_0 \mbox{d}(F_0+F_1)=\int f_0 \mbox{d}F_0+\int f_0 \mbox{d} F_1=1$ should be true?

A measure $\mu$ dominates another measure $\nu$ whenever $\mu=0$ implies $\nu=0$. If I would like to take the integral of a measurable function $f_0$, say the density function of the probability ...
0
votes
1answer
32 views

Converge in measure implies converge a.e if $f_n$ are monotone

Let $\{f_n\}$ be monotone sequence of functions such that $f_n$ converges in measure to $f$. Is it true that $f_n$ converges to $f$ a.e$?$ I am sure it has a sub-sequence that converges to $f$ a.e. ...