Questions tagged [singular-measures]
Two measures are said to be singular w.r.t. each other if they are supported on disjoint sets.
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Could the definition of singular measures be weakened?
The question:
The standard definition of singular measures is given on Wikipedia:
Two positive measures $\mu$ and $\nu$ defined on a measurable space $(\Omega,\Sigma)$ are called singular if there ...
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Derivative(s) of Cantor Measure (Donald L. Cohn ch. 6.2, exercise 6.2.4, related to lemma 6.2.5)
First, context: I'm doing a course on measure/integration theory following the book by Donald L. Cohn. In section 6.2, he defines the (upper and lower) derivates of a finite Borel measure $\mu$ on $\...
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Clarification on mutual singularity of probability measures
Let $P_1$ and $P_2$ be two probability measures on a measurable space, $(\Omega, \mathcal{F})$. Then $P_1$ and $P_2$ are mutually singular (denoted $P_1 \perp P_2$) if there exists $A \in \mathcal{F}$ ...
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$\lambda$ charge, $\mu$ measure, $\lambda \bot \mu$ implies $\lambda^{+},\lambda^{-},|\lambda|\, \bot\, \mu$
Let $\lambda$ a charge and $\mu$ a measure in $(X,\mathcal{X})$, with $\lambda\,\bot\,\mu $. So, $\exists A,B\in\mathcal{X}$ such that $A\cap B=\emptyset$, $X=A\cup B$ and $\lambda(A)=\mu(B)=0.$
I'm ...
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Singular Borel measures that are regular and the quotient of it's derivative over itself.
Let $\lambda$ be a positive regular Borel measure such that $\lambda\perp m$ on $\mathbb{R}^n$. Let $E$ be Borel with positive $\lambda$ measure and on $E$
$$\limsup_{r\to 0^+}\lambda B(x,7r)/\lambda ...
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Support of a Measure (Singular to Lebesgue)
On $[0,1]$ let m be lebesgue measure and $\mu$ a positive borel measure with $\mu \perp m$. Show that there exists a measure $\nu$ such that
$\|\mu-\nu\|< \epsilon$, and $m(\text{support}(\nu)) = ...
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Does the Lebesgue measure on the segment $y=x$ represent this distribution?
Set $\Omega=(-1,1)^2$. Consider the following measure on $\Omega$:
$\mu(A)=m(A \cap L)$, where $L=\{ (x,x) \, | \, -1 < x < 1\}$ (the segment of the line $y=x$ in $\Omega$) , and $m$ is the ...
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Singular bivariate distributions
It is known that singular copulas with prescribed support do not have a density, i.e. $\frac{\partial^2 \mathbf { C } \left( u _ { 1 } , u _ { 2 } \right)}{\partial u_1 \partial u_2} = 0$. For example,...
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Singular continuous
I am trying to construct a probability measure which is absolute continuous, singular constinuous and discrete. How can I do? I have not been able to find any example os such a measure. Might you help ...
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Singular function that is Holder for all $\alpha<1$
I am asking for an example of a singular continuous function that is Holder for all $\alpha<1$.
We know that such function cannot be Lipschitz, otherwise it is absolutely continuous.
We also know ...
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property of singular measure with respect to Lebesgue measure
Question: Let $\mu$ be a finite positive Borel measure on $\mathbb{R}$ that is singular to Lebesgue measure. Show that
$$\lim_{r\to 0^+} \frac{\mu([x-r,x+r])}{2r}=+\infty$$
for $\mu$-almost every $x\...
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An example of a a convolution of singular distribution and a Gaussian distribution that has a 'simple' pdf
I am looking for a nontrivial example of a singular distribution that when convolved with a Gaussian distribution has a pdf of a 'simple' form.
I let 'simple' be something that you interpret ...
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Reverse type $1-1$ inequality
If $\mu \in \mathcal{M}(\mathbb{T})$ is nontrivial and singular with respect to lebesgue measure, then
$$|\{\theta \in \mathbb{T} : M\mu(\theta) >\lambda\}| \ge \frac C\lambda \|\mu\|$$
where $| \...
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$\mu := \sum_{n\le 1} a_n \mu_n$ and $\mu _n \perp \nu$ . Show $\mu \perp \nu$.
Let $(\mu_n)_{n\in \Bbb N}$ and $\nu$ be $\sigma$-finite measures on $(\Omega , \mathcal A)$ and $(a_n)_{n\in \Bbb N}$ such that $\mu := \sum_{n\le 1} a_n \mu_n $ is a signed measure. Further, there ...
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Cantor function behavior close to point $1$
I was reading about the Cantor function (The Devil's Staircase) here and based on the picture and the description of the function, it seems to me that when $x \rightarrow 1$,
$$1-c(x)=o(1-x).$$
Is ...
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Understanding Folland's definition of two complex measures being mutually singular
In Folland's "Real Analysis", two complex measures $\nu=\nu_r + i\nu_i$ and $\mu=\mu_r + i\mu_i$ are said to be mutually singular (in symbols: $\nu\perp \mu$) if "$\nu_a \perp \mu_b$ for $a,b = r,i$". ...
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Singular measures
I'm given a measurable space $(\Omega, \mathcal F)$ and two probability measures $\mu,\nu$ on it with $\mu\ne\nu$. Consider the product measures $P:=⊗_{k\in \Bbb N}\mu$ and $Q:=⊗_{k\in \Bbb N}\nu$ on $...
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Properties of mutual singular measures
Let $\mu$ be a positive measure and $\nu_1, \nu_2$ arbitrary measures, all defined on the same measurable space $(S,\Sigma)$. We say that two arbitrary measure $\mu, \nu$ are mutually singular (...
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Measure theory (Singularity)
If $μ$ and $v$ be positive measures on $(X,Σ)$ such that each positive $ϵ$ there is a set $A$ in $Σ$ that satisfies $μ(A)<ϵ$ and $ν(A^c)<ϵ$. Now how to prove that $μ$ and $v$ are mutually ...
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Singular Measures on product $L^2$ Space
Let $<\Omega,\mathfrak{F},\mathbb{P}>$ be a probability space and let $\mathfrak{F}_1\cup\mathfrak{F}_2 =\mathfrak{F}$ be independent $\sigma$-fields.
Then do there exist singular measures $\...
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Singular measure example?
Can anyone tell me if I am grasping this correctly? Let $(X,\mathcal{F})$ be a measurable space where $X=[0,4\pi]$and then let $\lambda_{1}$ be a signed measure defined by
$$
\lambda_{1}(E)=\int_{E} ...
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Poisson Process independent Wiener Process using singular measures
I was reading some stochastic calculus of Jump processes and saw the following result:
If $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes ...
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Measures which are constant when not zero
Let $S$ be a finite set. I want to consider measures $\mu$ on $S$ which are constant only when not zero.
As an example, let $S$ be $\{a,b,c,d,e\}$, and take the measure: $\mu(a)=\mu(b)=\mu(e)=1/4,\...
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Does mutual singularity of measures imply absolute continuity?
I know that, a measure $\nu$ is absolutely continuous with respect to another measure $\mu$ if $\mu(A)=0$ implies $\nu(A)=0$. and the relation is indicated by $\nu<<\mu$.
On the other hand, ...
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$λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$, give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and $λ << ρ$
Le $f,g : \mathbb{R} → \mathbb{R}$ be extended integrable functions. Let $λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$. Give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and necessary ...
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Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.
Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
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Singularity of measures
I study the book of Gamelin about uniform algebras. As a corollary of the following Lemma: "let $K$ be a (weak star) compact convex set consisting of positive measures and a (complex) measure $\mu$ is ...
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Under the Borel measure associated to the Cantor function each of the intervals remaining in the construction of the Cantor set has measure $2 ^{-n}$
Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
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Decomposition of a measure
Let $\mu$ be the Lebesgue-Stieltjes measure on $\mathbb{R}$ corresponding to the distribution function, $F$ where
$$F(x) = \left\lbrace \begin{array}{ll}
0& \text{if} \,\, x<0\\
...
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two mutually singular measures witch both support is whole $\mathbf{R}$
I have some problem with an exercise(for homework):
Find two mutually singular measures $u$ and $v$ (Borel finite on $\mathbf{R}$) with $$\mathrm{supp}(u)=\mathrm{supp}(v)=\mathbf{R}.$$
I tried to ...
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How to calculate expectation of Cantor distribution without using p = 0.5
The question goes like this: given F(x) is the distribution function of Cantor distribution, how to calculate $E(x) = \int xdF(x)$ without using the argument of the following:
X is characterized by ...
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Consider two singular measures $ m$ and $v$ and $v$ is absolutely continuous with respect to $m$ show that $v=0$
Consider two singular measures $m$ and $v$ on a measurable space $(X,\mathcal{A})$ and $v$ is absolutely continuous with respect to $m$, i.e., $(v<<m)$. Show that $v=0$.
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Singular Lebesgue-Stieltjes measure
I read that if $F:\mathbb{R}\to\mathbb{R}$ is a non-decreasing singular function, i.e. a non-decreasing continuous function such that $F′(x)=0$ almost everywhere, then the Lebesgue-Stieltjes measure $\...
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Representations and mutually singular measures
I'm finding some difficulties with an exercise from Conway and I ask for some help in understanding it:
"Let X be a compact space and let $\{\mu_n\}$ be a sequence of measures in X.
For each $n$ let $...
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Do there exist two singular measures whose convolution is absolutely continuous?
Let $\mu, \nu$ be finite complex measures with compact supports on the real line, and assume that they are singular with respect to the Lebesgue measure. Can their convolution $\mu\ast\nu$ have a ...
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Two Radon measures and mutual singularity
Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$. Show that $\mu$ and $\lambda$ are mutually singular iff $D(\mu,\lambda,x)=\infty$ for $\mu$ almost all $x \in \mathbb{R^n}$.
I have looked ...
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How we compute expectation of a singular random variable?
In probability (or measure) courses, we often see the Cantor distribution that is singular with respect to the Lebesgue measure. Its CDF is increasing but whenever its differentiable, the ...
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Jordan decomposition theorem (question to singularity)
Let $(\Omega,\mathcal{A})$ be a measurable space.
(Hahn Decomposition Theorem) Let $\varphi\colon\mathcal{A}\to\mathbb{R}$ be a signed measure. Then there exist disjoint sets $\Omega^+\in\...
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Singular continuous functions
A function $f:[0,1]\rightarrow\mathbb{R}$ is called singular continuous, if it is nonconstant, nondecreasing, continuous and $f^\prime(t)=0$ whereever the derivative exists.
Let $f$ be a singular ...
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Question about singular continuous functions
A function $f:[0,1]\rightarrow\mathbb{R}$ is called singular continuous, if it is nonconstant, nondecreasing, continuous and $f^\prime(t)=0$ for almost every $t\in(0,1)$.
Let $f$ be a singular ...
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Product of a singular and continuous measures
Suppose we have a function $f(x,y)\geq 0$ and we integrate it with respect to a measure $\mu=\mu_{\mathrm{a.c.}}+\mu_{\mathrm{disc}}+\mu_{\mathrm{s.c}}$. Is it true that
\begin{equation}
\iint f(x,y)d\...
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Singular measures on Real line
Can some one please give me an example of continuous Singular measure on Reals which is not absolutely continuous to cantor-type sets. Thank you.
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CDF of non-atomic singular measure
Suppose $\mu$ is a non-atomic measure on the Borel subsets of $[0, 1]$ such that $\mu$ and
Lebesgue measure are mutually singular. Show that if $F$ is the cumulative distribution function of $\mu$, ...
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Singular measure with respect to translates
Let $\mu$ be some Borel measure on $\mathbf{R}$ such that, for every $t \neq 0$, the push-forward $(\tau_t)_* \mu$ is singular with respect to $\mu$ (where $\tau_t(x)=x+t$). What can we say about $\mu$...
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Singular measure with respect to Lebesgue measure
Let $\mu$ be a finite positive Borel measure on $\mathbb{T}$ such that $\mu$ is singular with respect to Lebesgue measure. Let $E$ be a closed subset of $\mathbb{T}$ such that $\mu(\{x\})=0$ for every ...
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Singular measure is regular
In Rudin Real and Complex Analysis, the proof of theorem 7.13 (page 142 3rd ed), why is $\mu$ regular?
If I understand correctly: prove that a singular complex Borel measure is regular. The reasoning ...
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Uniqueness of singular measure for inner function
A singular inner function $M$ (an analytic function on the open unit disk without zeros which takes on unimodular boundary values almost everywhere) can be written as
$$M(z)=c \exp\left(\int_0^{2\pi} \...
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Bound on the growth of a singular measure
Working through some measure theory, I came upon the Lebesgue-like decomposition for monotonic functions. In that context, I've cooked up a singular measure $\nu$ on $[0,1]$ about which I know only ...
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Mutually Singular measures
c.f. Rudin's Real and Complex Analysis (Third Edition 1987) Chapter 6 Q9
Suppose that $\{g_n\}$ is a sequence of positive continuous functions on $I=[0,1]$, $\mu$ is a positive Borel measure on $I$, $...
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Singular measures - approximate characteristic function
One can decompose a $\sigma$-finite measure $\mu$ on $\mathbb{R}$ in three parts:
$\mu_{ac}$: absolutely continuous
$\mu_{sc}$: singular continuous
$\mu_{pp}$: pure point
A common example for a ...
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