Tagged Questions

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

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17 views

Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let's even assume it is metrizable). A strictly positive measure on $X$ is the that gives positive measure to any non-empty open subset of $X$. Under which condition ...
1
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1answer
18 views

Measurable set contains a sequence

I found this question and didn't manage to extrapolate from the hint, could anyone help? Here's the question for the sake of completeness: Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: ...
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0answers
20 views

Showing that $\lim_n \sum_{k = 0}^n a_{kn} \lambda_1(A_{kn}) = \int_a^b f d\lambda$.

I need some help with the following: (i) Show that $\chi_{\mathbb Q \cap [a,b]}$ is measurable with respect to $\mathcal B(\mathbb R)|_{[a,b]}$ and compute $\int \chi_{\mathbb Q \cap [a,b]} ...
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1answer
26 views

Measure-theoretic analog of homeomorphism and isometry

If $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces and $f:X\to Y$ is a continuous bijective function between them such that $f^{-1}$ is also continuous, then the two topological spaces are said ...
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1answer
19 views

Equivalency in the elementary measure theory

Show that: $f\geqslant0$ and $\int f =0 $ $\Leftrightarrow$ $\mu$({$x$$\in$$X:$ $f($x$)>0$})=$0$ My idea: Let {$x$$\in$$X:$ $f($x$)>n$}=$E_{n}$ $\mu$({$x$$\in$$X:$ ...
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0answers
10 views

Cauchy's Functional Equation is satisfied by the paths of Levy processes (in distribution)

Consider Cauchy's Functional Equation: $$f(t+s)=f(t)+f(s).$$ Consider the following definition of a Levy process: A Levy Process on $\mathbb{R}^N$ is a $D([0,\infty);\mathbb{R}^N)$-valued random ...
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2answers
23 views

Can the integral get small outside a set with finite measure?

Let $(X, \mathcal{A}, \mu )$ be a measure space and let $ f : X \rightarrow \mathbb{\overline{R}}$ integrable. I just proved the fact that for every $\epsilon > 0$ we find $\delta > 0$ so that ...
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0answers
10 views

Continuity from below?

Let $(\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2))$ be our space. I have the set of "arc segments", $A(\theta,\eta,r,R)$, where $0\le\theta\le\eta<2\pi, \text{and } 0<r\le R$. The elements in the ...
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0answers
20 views

how that T is ergodic if and only if the only eigenfunctions $f \in L^2(\mu)$ of $U_T$ corresponding to the eigenvalue $1$ are constant functions.

Let $T:X \rightarrow X$ be a measure-preserving transformation. Assume that $(X,\mathcal{B},\mu)$ is a probability space. Show that T is ergodic if and only if the only eigenfunctions $f \in ...
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0answers
19 views

Integral inequality norm

I have read these one. It should be simple, but i can't find any answer: Let $D\subset \mathbb{R}^N$ measureable and bounded, and let $G:D\rightarrow\mathbb{C}^{N\times K}$ be measureable. Then, the ...
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0answers
13 views

Borel set's incompleteness

I was told that while M, class of all Lebesque measurable set is complete, sigma field generated by all open set B is not. Can someone give me an example of interval which is inside M yet is not ...
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1answer
21 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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1answer
21 views

Showing that Lebesgue measure is preserved by translations of the $d$-dimensional torus

Let $\underline{\alpha}=(\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$. Show that the transformation $R_{\underline{\alpha}}=\mathbb{T}^d \rightarrow \mathbb{T}^d$ defined by ...
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1answer
24 views

For each $\epsilon >0$ there is a $\delta >0$ such that if $\mu(E)<\delta$ then $\int_E |f_n|d\mu<\epsilon$ for all $n$.

Suppose $(f_n)$ is a Cauchy sequence in $L^1(X,\Sigma,\mu)$ where $(X,\Sigma,\mu)$ is a measure space. Prove that for each $\epsilon >0$ there is a $\delta >0$ such that if $\mu(E)<\delta$ ...
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0answers
17 views

Prove $\mathcal{B}(S^1) = S^1 \cap \mathcal{B}(\mathbb{R}^2)$

Let $S^1 = \{ (x,y) \in \mathbb{R}^2: x^2 + y^2 = 1 \}$. I am asked to prove that $$\mathcal{B}(S^1) = S^1 \cap \mathcal{B}(\mathbb{R}^2)$$ But I am unclear on what the sets actually are. Is ...
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1answer
43 views

Decompose a Riesz measure with atom into atom free and sum of Dirac measure

Let $\mu$ be a Riesz measure with some atom on $\mathbb{R}^n$. Show there exists two Riesz measure $\lambda$ and $\rho$ satisfying $\mu=\lambda+\rho$ where $\lambda$ has no atoms and ...
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1answer
17 views

Relation between two measures and absolute continuous

Let $\mu$ and $\rho$ be two measures on the same $\sigma$-algebra. Define $$\mathcal{E}=\{E|\mu(E^c)=0, E \quad\text{measurable}\}$$ and set $\gamma=\inf_{E\in\mathcal{E}}\rho(E)$. a) Show ...
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1answer
30 views

Example for $L^p$ and $L^q$ on $\mathbb{R^n}$

For $1\le p<q<\infty$, show that neither $L^p(\mathbb{R^n})\subset L^q(\mathbb{R^n}) $ nor $L^q(\mathbb{R^n})\subset L^p(\mathbb{R^n})$. Find a measurable function that belongs to ...
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3answers
62 views

Measure theory, measurable function, constant almost everywhere

Let f be a measurable function on the line such that for every real number x, we have $$f(x) = f(x+1) = f(x+\sqrt{2})$$. Show that f is constant almost everywhere. I am looking for a hit to start ...
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2answers
17 views

Show that for any $\mu$-measurable set $E$ we have $\mu(A\cap E) = \mu(B\cap E)$

Let $\mu$ be an outer measure on $X$. Let $A\subset X$ and assume there is a $\mu$-measurable set $B\supset A$ with $\mu(B)=\mu(A) <\infty$. Show that for any $\mu$-measurable set $E$ we have ...
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1answer
25 views

Caratheodory: Alternative Definition

Idea My idea is to facilitate Caratheodory's construction by composing it with Hahn-Kolmogorov. Problem Given a premeasure on a ring $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. Do the ...
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1answer
21 views

If $x$ is a Lebesgue point of $f$, $f \in L^p$ and $f(x)=0$, then it is a Lebesgue point of $f^p$ where $p>1$ (finite)?

If $x$ is a Lebesgue point of $f$ and $f \in L^p$, then it is a Lebesgue point of $f^p$ where $p>1$ (finite)? Here I use the following definition with ball replaced by cube. And I know if a ...
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1answer
15 views

Examples: Non-piecewise-constant + Non-Measurable

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded + Countably-Infinitely-Valued + ...
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0answers
27 views

Show that the 2-d skew product preserves the Lebesgue measure

Let $X=\mathbb{T}^2$. The $2$-$d$ torus and $\lambda$ the Lebesgue measure. Let $\alpha \in \mathbb{R}$ and consider the following map: $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2 : T(x,y) = (x ...
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3answers
37 views

$\frac{1}{x}\int_0^x f(t)dt=cf(x),\forall{x\in(0,\infty)}$ $\Rightarrow$ $f=0$ almost everywhere?

Suppose $f$ is a measurable real-valued function on $(0,\infty)$ such that $f\geq 0$ and $$\frac{1}{x}\int_0^x f(t)dt=cf(x),\quad\forall{x\in(0,\infty)}$$ for some constant $c$. Does it follow that ...
3
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0answers
31 views

Hypothesis in Lusin's Theorem

If $f\colon E\rightarrow \mathbb{R}$ is measurable function, and $m(E)<\infty$, then by Lusin's theorem, restriction of $f$ to a large closed set is continuous. It is in the proof, I saw, that ...
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0answers
35 views

Show that a measure is a probability measure

I have trouble with this question: We define an arc segment $B(\theta, \eta, r, R)=\{x \in \mathbb{R}^2\vert \omega(x)\in [\theta,\eta], \Vert x \Vert_2 \in [r,R] \}$ where $0 \leq \theta \leq \eta ...
2
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1answer
37 views

Show that Fourier transformation is differentiable if $\int|xf(x)|\,d\lambda<\infty$

Let $f\in\mathcal{L}^1(\mathbb{R},\mathcal{M},\lambda)$. Then we define the Fourier transform of $f$, denoted $\hat{f}$, by ...
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1answer
25 views

Prove that a subset is measurable is and only if the measurable of the set equal to the sum of that subset and its complement

Let $X$ be a set and $\mathscr{A}$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a measure on $\mathscr{A}$ such that $l(X) < \infty$. Define $\mu^{*} $ as $$ ...
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1answer
41 views
+50

Show that a set defined on a arc segment is closed

A arch measure defined by $B(r,R,\eta,\theta)$ defined by two angels $0<\eta<\theta<2\pi$ and 2 radians $R>r>0$. The arc segment consists of 2 vectors $x\in \Bbb R^2$, which has the ...
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3answers
84 views

Apply dominated convergence theorem to show differentiability

Let $f,g \in L^p(\mu), 1 < p < \infty$. Show that the function $$\phi(t)=\int |f+tg|^pd\mu$$ be differentiable at $t=0$ and find $\phi'(0)$. My try, $\psi(t)=|f+tg|^p$ is differentiable at ...
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2answers
23 views

Sequence of Measurable Sets

Suppose that $ E $ is a measurable subset of $ R $. Suppose $ \left \{ E_{i} \right \} $ is sequence of measurable subsets of $ E $. For any $ x \in E $, there exist an $ N_{x} $ ...
3
votes
1answer
38 views

A function defined by $L^p$ integral is continuous on the boundary

Suppose $f$ is a measurable function on $X$, $\mu$ is a positive measure on $X$, and $$g(p)=\int_X|f|^p d\mu=||f||_p^p, (0<p<\infty)$$ Let $E=\{p|g(p)<\infty\}$. Assume $||f||_\infty >0$. ...
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1answer
18 views

Set of differences of a set $A$ where $\lambda(A) > 1$

Here's the problem ($\lambda$ denotes the Lebesgue measure): Let $A \subset R^2$ with $\lambda(A)>1$. Prove: $(A-A) \cap \mathbb{N}^2 \ne \emptyset$ After trying to prove it for sets $A ...
3
votes
0answers
23 views

Integration with respect to signed measure and Radon Nikodym theorem

I aim to show the following question: Let $\mu$ be a $\sigma$-finite measure, and $\lambda$ a finite signed measure on $(X,M)$ satisfying $\lambda\ll\mu$, let $h=\frac{d\lambda}{d\mu}\in ...
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8answers
2k views

What properties are true for “almost all real numbers”? [on hold]

Over the years, I remember hearing of a lot of proofs of properties that applied to "almost all real numbers", that is the exceptions to the rules had measure zero. But I can't remember many of them. ...
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0answers
35 views

relation of these two continuity

Suppose that (1)$X=\mathbb{R}^n$. (2)$ M=\{ U$| $U$is mesurable subset of $X\}$. (3)$f:X\rightarrow X$ induces $f':M\rightarrow M$ s.t. $f'(U)=f(U)$ for all $U \in M$. (4)for $U\subset X$, $C(U)$ is ...
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0answers
20 views

I want to cut an A4 sheet into 3, 2 at the top corners and 1 in bottom centre [on hold]

i want to cut an A4 sheet into 3 pieces but not the normal way. 1 at each top corner and 1 in bottom centre each the same size? any ideas?
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1answer
30 views
0
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0answers
20 views

Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aime is to understand the usual augmentation filtration. More pricesely, I want to understand the last identity in this PDF file. http://onlinelibrary.wiley.com/doi/10.1002/0470863617.app1/pdf ...
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0answers
25 views

Closed set through continuity

I have the measure space $(\mathbb{R}^2,\mathcal{B}(\mathbb{R^2}))$ and the set $A=\{x \in \mathbb{R}^2\mid w(x)\in[\theta, \eta], ||x||_2\in[r,R]\}$, where $0\le\theta\le\eta<2\pi, \text{and } ...
0
votes
1answer
18 views

Completion of Borel Algebra with point mass measure at 0.

Trying to find the complition of $(\mathbb{R},\cal{B}(\mathbb{R}),\mu_0)$ where $\cal{B}(\mathbb{R})$ is Borel Algebra and $\mu_0(E)=1$ if $0\in E$ and zero otherwise. So I know I need to include the ...
0
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2answers
26 views

Translating expected values between two sets of related iid variables

The setting: $\mu$ is a probability measure on $\mathbb{R}$, $f: \mathbb{R} \to [0, \infty)$ so that $0 < ||f||_{L^1(\mu)} < \infty$, and $v$ is another probability measure defined by $v(A) = ...
2
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1answer
73 views

The measure generated by the Cantor staircase and the intersection of the Cantor set with its translate

Suppose that $T$ is the shift $\bmod 1$ of the Cantor set by an irrational number $\alpha\in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. I'd like to ...
2
votes
2answers
39 views

Assume that $N\subset \mathbb{R}$ is not Lebesgue measurable. Is then the set $ N\times \mathbb{R} $ also not Lebesgue measurable?

Assume that $N\subset \mathbb{R}$ is not Lebesgue measurable. Is then the subset $ N\times \mathbb{R} \subset \mathbb{R}^2 $ also not Lebesgue measurable? Im not sure whether this is true or not. The ...
1
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1answer
27 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
0
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1answer
17 views

Prove that $\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$

Do we have the following identity? $$\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$$ Here $C_i$ is a subset of a set $\Omega$.
2
votes
1answer
31 views

Integration with respect to a signed measure

Let $\mu$ be a singed measure, $f\in C_c(X)$, I want to show $$\int fd(c\mu) = c \int fd\mu, \forall c \in \mathbb{R}$$ Since $c\mu$ is also a singed measure, I think by definitionm I need to show ...
1
vote
1answer
23 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
0
votes
1answer
25 views

$\varphi(x)=\int_{[\xi_0,\xi]}f(x+t)d\mu_t$ absolutely continuous and summable on $\mathbb{R}$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$. I read that the function$$\varphi(x)=\int_{[\xi_0,\xi]}f(x+t)d\mu_t$$is absolutely continuous on any real closed ...