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

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2
votes
1answer
34 views

Prove that $\sum_{k=0}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \mathrm{ d}x$

Problem Statement Prove that $$\sum_{k=1}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \ \mathrm{d}x$$ Background I just learned the limit theorems (MCT, LDCT, Fatou's Lemma). This ...
2
votes
0answers
40 views

Inner measure (inner set function) on functional closed sets

I'm struggling with the following problem: Let $X$ be a set and $\mathcal{Z}:=\{Z\subseteq X \,\big|\,\exists\,\psi\in\mathcal{C}(X)\,:\,Z=\psi^{-1}(\{0\})\}$ the family of functional closed sets ...
1
vote
1answer
33 views

The Cantor set and ternary expansions

I'm trying to prove that the Cantor set $\mathcal{C}$ contains all numbers $x \in [0,1]$ with ternary expansion $x = \sum_{k=1}^\infty \frac{a_k}{3^k}$, such that $a_k=0$ or $a_k=2$. I'm going by ...
0
votes
1answer
18 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
2
votes
1answer
61 views

$\sigma+$-field of a Brownian motion

For a standard Brownian motion define \begin{align}\mathcal{F}_{0+} &= \bigcap_{t>0} \mathcal{F_t},\\ \mathcal{F_t} &= \sigma(W_s, 0 \le s \le t)\end{align} Which of the following ...
-1
votes
1answer
28 views

Semi-metric defining convergence in measure

Let $(X, \mu)$ be a measure space such that $\mu(X) \lt \infty$. Let $F$ be a separable metric space. A function $f: X\rightarrow F$ is said to be mesurable if $f^{-1}(U)$ is measurable for every open ...
3
votes
3answers
54 views

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

Background I seem to be having issues recognizing valid bounding functions when applying the Lebesgue Dominated convergence theorem. Here are two examples I did that I do not think are justified. ...
1
vote
0answers
29 views

Evaluating an integral of a function $a:\mathbb{N}\to [0,\infty)$ with respect to certain measure

Let $(p_n)$ be a sequence of positive reals for $n\ge 0$. For any $N\subset \mathbb{N}$, define $\mu(N)=\sum_{n\in N} p_n$, with the convention that $\sum_{n\in \emptyset} p_n=0$. Then $\mu$ is a ...
4
votes
2answers
63 views

Composition of almost everywhere continuous functions $\mathbb{R}\rightarrow\mathbb{R}$

If $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are continuous it is well known that the composition $f\circ g$ is also continuous. But what happens if we assume that $f,g$ are only almost everywhere ...
1
vote
1answer
32 views

What's an example where this definition of measure has a limit that does not exist?

I'm looking for an example. Consider defining the measure of a set $E \subseteq \Bbb{R}^d$ by a limit: Take $$m(E) := \lim_{N \to \infty} \frac{1}{N^d} \cdot \left| E \cap \frac{1}{N} \Bbb{Z}^d ...
2
votes
1answer
44 views

Find the limit $\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$

Problem Statement Find the limit $$\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$$ Attempt This problem is very similar to the following and I am basically going to ...
1
vote
1answer
31 views

“measure zero” and “measurable function” on Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold (which doesn't have to be orientable). As far as I know, the metric $g$ induces a "canonical" measure $\mu$ and so one can talk about sets $U\subset M$ of measure ...
1
vote
0answers
23 views

Construction of the Lebesgue measure

Is it possible to construct the Lebesgue measure on an $\underline{\text{abstract}}$ Boolean algebra? Or, does the greater abstraction of considering Boolean algebras instead of fields of sets ...
0
votes
3answers
26 views

Equivalent way of writing the norm of Lp

Given a measurable $E\subset \Bbb R^d $ and a measurable function $f:E\rightarrow \Bbb R^d $, prove that : $$ \int (\left\lvert f \right\rvert)^r d\mu = r\int_{0}^\infty t^{r-1} \mu(\{x \in E ...
0
votes
1answer
33 views

If $t \mapsto \mu((-\infty, t))$ is differentiable at $x$, and $a_n \nearrow x$, does $\mu(\{a_n\})/(x-a_n)$ tend to zero?

Let $\mu$ be a complex (and therefore finite) Borel measure on $\mathbb{R}$ and suppose we have a function $f(t)=\mu((-\infty,t))$. Further suppose that $f$ is differentiable at a point ...
0
votes
0answers
9 views

Prove that $\int_{\mathbb R^n} u_\epsilon \, divT=\int_F div(T*\rho_\epsilon). $

Let $F\subseteq \mathbb R^n$ be a set and let $\chi_F$ its indicator function, regularized with functions $\rho_\epsilon\in C^\infty_c(\mathbb R^n)$ such that $u_\epsilon:=\chi_F* \rho_\epsilon\to 0$ ...
1
vote
1answer
36 views

Folland's Real Analysis Exercise 1.22a

The exercise states: Let $(X, \mathcal{M}, \mu)$ be a measure space, $\mu^*$ the outer measure induced by $\mu$ according to (1.12), $\mathcal{M}^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, ...
0
votes
0answers
4 views

Jacobian for bi-Lipschitz mappings between Hilbert cubes

It is well known that for every bi-Lipschitz function $f:M\to N$ between finite-dimensional smooth manifolds has a Jacobian, i.e. there exists $J_f:M\to \mathbb{R_{\geq 0}}$ such that for every ...
1
vote
1answer
26 views

Relations between the topologies of $L^0$ and $L^1$ on finite measure space

Let $(\Omega, \mathcal F, P)$ be a probability space and denote by $L^0(\Omega, \mathcal F, P)$ the space of all random variables $X : \Omega \to \mathbb R$ (i.e. measurable functions between ...
0
votes
0answers
15 views

characeterization of zero sets of the riemannian measure of a riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold (does not have to be orientiable). Then there exists the Riemannian measure $\nu(g)$ on $M$. Let $(U_i,x_i)$ be a finite covering of $M$ of charts and let ...
0
votes
0answers
47 views

If $f_n$ is integrable and non-negative, $f_n \rightarrow f$ a.e, $\int f_n \rightarrow \int f$. prove $\int_A f_n \rightarrow \int_A f$

Problem Statement Suppose $(X,\mathcal{A},\mu)$ is a measure space, each $f_n$ is integrable and non-negative, $f_n \rightarrow f$ a.e, and $\int f_n \rightarrow \int f$. prove that for each $A \in ...
1
vote
1answer
20 views

Proving that a function is measurable

Let $(X,\mathcal{A},\mu)$ be a measure space and $(Y,d)$ a separable metric space. Given $f,g:X\to Y$ $\mu$-measurable functions, prove that $h:X\to \mathbb{R}$, defined by $x\mapsto d(f(x),g(x))$ is ...
2
votes
0answers
29 views

Brownian motion, find minimum of function

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \geq 0)$ a Brownian motion and $(\mathcal{F}(t),t \geq 0)$ its natural filtration. Suppose $0 \leq s \leq t$ and let $f:\mathbb{R} ...
0
votes
0answers
32 views

Measure Theory by Halmos, theorem B, page 22

This is an excerpt from Halmos' "Measure Theory" at page 22 and 23. This proof seems wrong to me. I could not prove that the class of all finite union of elements of $E$ is a ring of sets. I can only ...
5
votes
2answers
111 views

A construction of sigma-algebras - surely not new, right?

I know no descriptive set theory. I've stumbled on something that must be well known, being so simple. But it contradicts something I've been told by smart people; the question is whether it's well ...
0
votes
0answers
31 views

Show $Z$ as a function defined on $\Omega$

I have a random variable $X(n)$, with $n=0,1,2$, (for example $X$ could represent stock price movements in a binomial tree) built on the probability space $\Omega$ with Lebesgue measure $(n=1,2)$. ...
0
votes
0answers
15 views

counter example on union of sigma-fields [duplicate]

Let $\{F_n\}$ be a sequence of $\sigma$-fields in $\Omega$ such that $F_n\subset F_{n+1}$. Find an example such that $\bigcup_{1}^{\infty} F_n$ is not a $\sigma$-field. Can any one help me with this ...
2
votes
1answer
41 views

Borel Measures: Regularity

Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{R}_+:\quad\mu(\mathbb{C})<\infty$$ Then it is regular: $$\mu(A)=\sup_{A\supseteq ...
1
vote
2answers
25 views

How to prove limit of measurable functions is measurable

I need help to prove the following theorem Suppose $f$ is the pointwise limit of a sequence of $f_n$, $n = 1, 2, \cdots$, where $f_n$ is a Borel measurable function on $X$. Then $f$ is Borel ...
-1
votes
0answers
16 views

upper and lower semi continuouis

let $X$ be a topological space,$ f_n:\mathbf X \to \mathbb R$ is sequence of lower semi continuous functions then the $ \sup \{f_n\}=f $ is also lower semi continuous proof: f is ...
2
votes
1answer
43 views

If $X$ has a density, then does $Y:=g(X)\cdot 1_{\{X>0\}}$? (No…?)

I think there is a typo in my probability theory book's exercises. Probability Essentials by Jacod and Protter, exercise 11.9 says: Let $X$ have a density, and let $$Y=ce^{-\alpha ...
1
vote
1answer
23 views

For an infinite set, prove that the collection of subsets A such that A is finite, or A complement is finite is a sigma algebra

This problem seems incorrect to me. I proved it for the case where we replace finite with countable, but I don't believe it is true here - a sigma algebra must be closed under countable union - let ...
0
votes
2answers
41 views

If $f$ is non-negative and measureable, prove that $\lim_{n \to \infty} \int \min(f,n) \rightarrow \int f$.

Problem Statement Let $f$ be a non-negative measurable function. Prove that $$\lim_{n \to \infty} \int \min(f,n) \rightarrow \int f$$. Attempt First, if $f= \infty$ on a set of positive measure, ...
0
votes
1answer
36 views

Equality of these two sigma algebras?

Let $\mathbb{B}$ denote the set of Borel sets of $\mathbb{R}$. Let $\Omega = [0,1]$. Let $A_1 = \sigma \text{(open subsets of }\Omega)$, that is, sigma algebra generated by open subset of $\Omega$. ...
0
votes
1answer
53 views

Complex Measures: Polynomials

Given the complex plane $\mathbb{C}$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\operatorname{supp}\mu\subseteq\overline{B_r}$$ Then one has: ...
3
votes
1answer
28 views

Convergence in probability, in the sense of weak convergence of measures

I am reading a paper where the author has a family $(\rho_t : t \geq 0)$ of random probability measures (on the real line with Borel sigma-algebra), and a measure $\rho$. One of his theorems says that ...
0
votes
1answer
39 views

Spectral Measures: Constructions

Any constructions welcome!!! Given a Hilbert space $\mathcal{H}$. Regard spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ That are additive: ...
0
votes
2answers
25 views

Borel Measures: Single-Valued

Given the complex plane $\mathbb{C}$. Consider the Dirac measure: $$\mu_\lambda(A):=\chi_A(\lambda)$$ Then it attains only zero and one: $$\mu_\lambda(A)=0,1$$ Are there any other such measures?
2
votes
1answer
23 views

Any relationship between Hausdorff measures

Let $ S_1= ( [0,1], d_1 ) $ and $ S_2 = ( [0,1], d_2 ) $ be two metric spaces, where $ d_1 = |x - y|$ and $d_2 = (1/2^i) $ where binary expansion of x and y matches upto $ i^{th} $ coordinate. Let $ ...
0
votes
0answers
24 views

Completeness of a certain semi-normed space whose semi-norm is defined by an upper integral

Let $(X, \mathcal M , \mu)$ be a measure space where $\mathcal M$ is a $\sigma$-algebra on a set $X$ and $\mu$ is a measure defined on $\mathcal M$. Let $B$ be a Banach space over $\mathbb R$ or ...
0
votes
1answer
34 views

If $\mu$ is $\sigma$-finite, then there exist increasing simple functions $s_n \rightarrow f$ with $\mu(\{x:s_n \neq 0\})< \infty$

Problem If $\mu$ is $\sigma$-finite, $f$ non-negative and measurable, then there exist simple functions $s_n$ increasing to $f$ at each point such that $\mu(\{x:s_n \neq 0\})< \infty$ for each ...
1
vote
1answer
36 views

Proving continuity for all $x$.

I am having difficulty in proving the following problem. Any hints would be greatly appreciated. Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ be such that for each $y \in [0,1]$, $f(\cdot,y)$ is ...
0
votes
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
vote
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
vote
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
vote
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
votes
0answers
32 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
votes
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 ...