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

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0
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1answer
16 views

If $s=\sum_{i=1}^{m}a_i\chi_{A_i}=\sum_{j=1}^{n}b_j\chi_{B_j}$ then $\sum_{i=1}^{m}a_i\mu(A_i)=\sum_{j=1}^{n}b_j\mu(B_j)$

Background Let $\chi_A$ be the characteristic function of the set $A$. A simple function $s$ is a function of the form $$s(x)=\sum_{i=1}^{n}a_i\chi_{E_i}(x),$$ where $a_i \in \mathbb{R}$ and $E_i$ ...
0
votes
1answer
19 views

Giving a bound of the norm of a convolution

Let $f:(-1,1)\to \mathbb{R}$ be a smooth function with compact support. Suppose that $f(x)\geq 0$ for all $x$ and $\int f =1$. Extend $f$ to $\mathbb{R}$ by $f(x) = 0$ if $x\not \in (-1,1)$. Show that ...
1
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1answer
30 views

Lower Bound of Hausdorff Dimension of Cantor Set

Consider a Cantor set $E$ where the intervals at every level of the construction maintain a minimum spacing and have a finite number of intervals on each level. I have two questions regarding finding ...
3
votes
0answers
106 views

Nontrivial normed functional on the bounded functions from $\mathbb R^2$ into $\mathbb R$ invariant by isometries

I am trying to show that there exists a nontrivial normed functional on $\mathbb R^2$ invariant by isometries. That is: If $A$ is any set, let $\mathcal B_{A}=\{f: \mathbb R^2 \rightarrow \mathbb R: ...
1
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1answer
22 views

$S(\Omega \sqcap A)=S(\Omega)\sqcap A$ Halmos Measure Theory

I'm having trouble grasping the proof of theorem E, section 5, chapter 1 in Halmos' Measure Theory. Let $X$ be a nonempty set, and $\Omega$ a family of subsets of of $X$. Given $A\subset X$, denote ...
1
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1answer
83 views

Help me find $\mu(x)$ in $\int_0^1 x^n d \mu(x) = \lambda^n$

For the following integral $$\int_0^1 x^n \mathsf d \mu(x) = \lambda^n,$$ Where $\lambda$ is some constant with norm less than 1, and $\mu(x)$ is a Carleson measure in the Unit Disc. What candidate ...
0
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0answers
33 views

Triangular projection on kernels of trace class operators

Let $k$ be a kernel of a trace class integral operator on $L^2(0, 1)$, and assume that $k(x, y)=-k(y, x)$. Define $l(x, y)=k(x, y)$ whenever $x>y$, and $l(x, y)=0$ if $x<y$. Does the integral ...
2
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0answers
28 views

Weakly measurable functions

Let $(X, \mathcal M)$ be a measurable space, where $\mathcal M$ is a $\sigma$-algebra on a set $X$. Let $f\colon X \rightarrow B$ be a map where $B$ is a Banach space over $\mathbb K$ which is ...
2
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1answer
21 views

Composition of 2 Lebesgue measurable functions is not lebesgue measurable: Are these two functions Borel Measurable?

Background Let $(X,\mathcal{A})$ be a measure space and $\mathcal{B}$ the Borel $\sigma-$algebra. A function $f: (X,\mathbb{\mathcal{A}}) \rightarrow (\mathbb{R},\mathcal{B})$ is $\mathcal{A}$ ...
4
votes
1answer
58 views

What is so good about the $L^2$-norm of the second derivative being small?

One of the main properties of cubic splines is the minimality property which basically means that if $s$ (cubic spline) and $g$ (some other function) interpolate $f$ in a certain way then $$\Vert s'' ...
1
vote
2answers
35 views

Lebesgue measurable and non-measurable sets

Is it possible to construct such $A \subset [0,1]^2$ that is not Lebesgue measurable, but it's projections to coordinate axes are measurable? Similarly construct subset that is measurable, but ...
0
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0answers
10 views

Selfdecomposability and Lévy processes

I am trying to understand Levy processes and I have some issues with this. A random variable x is selfdecomposable then x has a representation of the form \begin{equation} x=\int ...
6
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1answer
43 views

A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable

Background Give an example of a collection of measurable non-negative functions $\{f_\alpha\}_{\alpha \in A}$ such that if $g$ is defined by $g(x)=\sup_{\alpha \in A} f_{\alpha}(x)$, then $g$ is ...
0
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0answers
12 views

Equivalence of definitions of measurable functions

Recently I read in a book that the definition $f^{-1}(E) \in X$ for every Borel set E is equivalent to the standard definition that f is measurable if for every real number $\alpha$ $\{x\in X: ...
-1
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1answer
42 views

Measure theory examples of non-empty sets with measure zero. Area of plane. [closed]

Give two examples of non-empty sets with measure zero, for some measure given in $\mathbb{R}^2.$
1
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1answer
21 views

How to show that consecutive hitting times are a sequence of stopping times?

Let $\{X_n:n=0,1,\ldots\}$ be a martingale with respect to a filtration $\{\mathcal F_n\}$. Let $A,B$ be nonempty, disjoint Borel sets and define $T_0=0$, \begin{align} S_n &= \inf\{m\geqslant ...
2
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1answer
30 views

monotonic linear functional on $C_+(X)$

Let $X$ be a compact metric space. Let $C_{+}(X)$ be the set of all continuous non negative functions on $X$. Let $\lambda : C_{+}(X) \to [0,\infty)$ such that ...
3
votes
2answers
91 views

Is the limit of measurable step functions measurable?

Let $(X, \mathcal M)$ be a measurable space where $\mathcal M$ is a $\sigma$-algebra on a set $X$. Let $f$ be a map $X \rightarrow E$ where $E$ is a metric space. Suppose $f$ is the pointwise limit of ...
2
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1answer
27 views

Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
4
votes
3answers
49 views

Measure of countable union

In my measure theory class, the professor prove that if $\mu$ is a finite measure on a space $X$ ($\mu(X) < \infty$) and $A_1 \subset A_2 \subset A_3 \subset \cdots$, then $\lim_{n\to\infty} ...
0
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1answer
19 views

If $f=g$ on $(x-r,x+r)\cap (0,1)$ and $g$ is Borel measurable, then $f$ is Borel measurable.

Question Let $f:(0,1) \rightarrow \mathbb{R}$ be such that for every $x\in (0,1)$, there exists $r>0$ and a Borel measurable function $g$, both depending on $x$, such that $f$ and $g$ agree on ...
1
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0answers
26 views

Induced probability measure for argmax

Let us consider a metric space $S$ consisting of random functions $f:[0,1]\rightarrow \mathbb{R^+}$ and a probability measure $P$, defined on this space. Now suppose we define a new function $$\hat ...
0
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0answers
8 views

$d \log f'$ is a compactly supported radon measure for $f\in C_c^{1}$ with bounded variation

In a note of John N. Mather, "Commutators of Diffeomorphisms, III: a group which is not perfect," he says that it is well known that if $f$ is a compactly supported $C^1$ diffeomorphism on ...
1
vote
1answer
39 views

Why isn't $\mathcal D$ a sigma-algebra?

I came across the statement that if $(\Omega, \mathcal F, \mathbb P)$ is a probability space and $E \in \mathcal F$ then $$\mathcal D := \{ A \in \mathcal F \mid A \text{ and } E \text{ are ...
0
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0answers
10 views

Purely nondeterministic weakly stationary processes

I found a necessary and sufficient condition for a stochastic process being purely nondeterministic in Ihara (1993). As follows: A weakly stationary process $X$ is purely non-deterministic if and ...
2
votes
1answer
33 views

Regularity of $\phi$ in order that $\int g_h \phi \,dx \to \phi(0)$

Define the sequence of functions $(g_h)_h$ where $$g_h(x):= h\, \chi_{[0,1/h]}(x)$$ and the sequence of measures $$(\mu_h(dx))_h:= g_h(x)\,dx.$$ We want to show that $\mu_h ...
0
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1answer
24 views

Find two functions in $L_p(\Bbb R)$, whose product $f\cdot g$ does not belong to $L_p(\Bbb R)$.

How can I find two functions in $L_p(\Bbb R)$, with their product $f\cdot g$ not belonging to $L_p(\Bbb R)$?
2
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1answer
47 views

Natural and completed natural filtration not right-continuous

I'm looking for an example of a stochastic process, such that the natural filtration and the completed natural filtration aren't right-continuous. I defined the process $(Z_t)_{t \geq 0}$ as $Z_t = t ...
2
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1answer
46 views

The sigma algebra generated by open-dense subsets of $\mathbb{R}$

What is the description of the sigma algebra generated by all open-dense subsets of $\mathbb{R}$? Is it equal to the Borel sigma algebra?If not how is the structure of this sigma algebra?
0
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1answer
34 views

Two positive measures are mutually singular iff their sum is the variation of their difference

Let $\mu$ and $\nu$ be two finitely positive measures on measurable space $\left ( X, \mathfrak{A} \right )$. Prove that $$\mu \perp \nu \Leftrightarrow \mu + \nu = \left | \mu -\nu \right |$$ ...
3
votes
1answer
32 views

Necessity of generalization of Dominated Convergence theorem

In Royden's Real Analysis there's this generalization of Lebesgue's Dominated Convergence theorem (p.92): Let $\{g_n\}$ be a sequence of integrable functions which converges a.e. to an ...
3
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0answers
31 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} ...
2
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1answer
36 views

Benefit from measure theory

With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral) What I know: With the Lebesgue integral we need less requirements to ...
3
votes
1answer
49 views

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: ...
2
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0answers
31 views

Prove that $\lim_{r \to 1} \int_{-\pi}^{\pi} f(re^{i\theta}) d\theta = \int_{-\pi}^{\pi} f(e^{i\theta}) d\theta$

...if $f$ is continuous in an open set $\Omega$ containing the unit circle $T$. Is the proof something along the line of: $T$ is compact hence $\exists \epsilon > 0$ such that $D(z;\epsilon) ...
1
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0answers
24 views

Video lectures on Measure and Integration

Does anyone know a good online lecture series on measure theory and Lebesgue integration? I looked at the MIT open courseware but I could find only lecture notes. I am interested in lectures on this ...
1
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1answer
30 views

If $f(x) \le f(Tx)$ then $f(x)=f(Tx)$ almost everywhere ( $T$ is $\mu$-invariant )

Let $X$ be a probability space with probability $\mu$. Let $T:X\to X$ be a measurable and $\mu$-invariant transformation, i.e $\mu \left(T^{-1}A \right) =\mu A. $ for each measurable subset $A\subset ...
3
votes
0answers
37 views

Lebesgue measure of a parallelepiped

Suppose we have $n$ linearly independent vectors $\mathbf{x}_1$, $\cdots$, $\mathbf{x}_n$ in $\mathbb{R}^n$. Let $\mathbf{X}$ be the $n \times n$ matrix with column $k$ given by $\mathbf{x}_k$, $k = ...
3
votes
2answers
49 views

How to demostrate that $\int_{\{a\}} f(x) d \mu= \mu(\{a\}) f(a)$?

I don´t know how to demostrate that: $$\int_{\{a\}} f(x) d \mu= \mu(\{a\}) f(a)$$ Note: I have read on a book that "The Lebesgue Integral of a constant function on a measurable set will be that ...
0
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1answer
42 views

Basic measure theory question

It is a very important question for me. If we define the measure ${\mu}$ like this: ${\mu}=\left\lbrace \begin{array}{l} {\mu} (\lbrace a \rbrace )>0 \\ {\mu} (\mathbb{R}-{\lbrace ...
0
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0answers
21 views

Borel regular measures equal if equal on open sets

Let $\mu$ and $\nu$ be Borel regular outer measures in a (separable metric) space X. We know that $\mu(U) = \nu(U)$ for every open set $U$. We wish to show that $\mu(A) = \nu(A)$ for every set $A ...
0
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0answers
33 views

Understanding what tightness means for a sequence of measures.

I am trying to understand the concept of tight measures. I have as a definition that $(\mu_n)_n$ is tight iff $\forall \epsilon>0, \exists K$ compact s.t. $\forall n : \mu_n(K)\geq ...
0
votes
2answers
28 views

Showing that the L1 norm of a given sequence of functions diverges

For $n=1,2,3,\ldots$ define $f_n:\mathbb R\to\mathbb R$ by $$f_n(x) = \frac{\sin(x)\sin(nx)}{x^2}.$$ Then certainly each $f_n$ is integrable on the real line. However, I have to show that the $L^1$ ...
0
votes
1answer
51 views

Jensen inequality for convex functions - infinite countable number of weights

Does Jensen inequality for convex functions hold if there is countable infinite number of weights? For example weights are given by sequence $(\frac{1}{2^n})_{n\in\mathbb{N}_1}$ ? If not, where is ...
1
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1answer
29 views

Itô's formula yields an Itô process

In our course on stochastic analysis, we proved the following version of the one-dimensional Itô formula: Let $\{W_t\}_{t\ge 0}$ be a one-dimensional Brownian motion w.r.t. some (right-continuous and ...
1
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1answer
15 views

Borel sigma field on inifnite (countable) set

If $\Omega$ is inifnite (countable) set, then borel sigma algebra on that set is equal to $2^\Omega$? What if $\Omega$ is uncountable?
0
votes
1answer
34 views

Linear Map is Homeomorphic in $\mathbb R^k$

I am having trouble understanding a proof in Rudin's "Real and Complex Analysis." The theorem states that To every linear transformation $T$ of $\mathbb R^k$ into $\mathbb R^k$ corresponds a real ...
0
votes
5answers
273 views

How can an open set be equal to a union of half-open sets?

In discussing $\sigma$-algebras I have seen the following used in proofs: $$(0,1)=\bigcup_{i=1}^\infty \left[ \frac{1}{n} , 1 \right)$$ In other words the open set is equal to a denumerable union of ...
0
votes
1answer
26 views

example for a function convergent in measure and in $L_p$ but not almost everywhere

I was looking for an example of a sequence $\{f_n\}_{n=1}^{\infty}$ such that $f_n\rightarrow f$ in measure and in $L_p$, but not almost everywhere. The book I'm studying (Real Analysis for Graduate ...
8
votes
1answer
299 views

Two sets $X,Y \subset [0,1]$ such that $X+Y=[0,2]$

A set $X\subset \mathbb{R}$ is called nice if, for every $\epsilon > 0$, there are a positive integer $k$ and some bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 ...