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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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 an 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 yourself. ...
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1answer
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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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1answer
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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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1answer
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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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355 views

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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1answer
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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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2answers
102 views

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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2answers
131 views

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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2answers
667 views

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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1answer
109 views

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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2answers
767 views

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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1answer
2k views

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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1answer
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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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120 views

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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2answers
465 views

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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1answer
891 views

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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384 views

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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1answer
113 views

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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1answer
128 views

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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1answer
341 views

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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1answer
456 views

Question on singular measures and absolute continuity

Below is a question from an old real analysis exam that I need help in solving. Let $\mu, \nu$, and $\lambda$ be $\sigma$-finite, nonnegative and nontrivial measures on the measure space $(X,\mathcal{...
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2answers
253 views

A sequence of singular measures converging weakly* to a continuous measure

Can anyone provide a sequence of singular (w.r.t. Lebesgue measure) measures $\in\mathcal{M}([0,1])=C[0,1]^*$ converging $weakly^*$ to an absolutely continuous (w.r.t. Lebesgue measure) measure?
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How is a singular continuous measure defined?

On a measurable space, how is a measure being singular continuous relative to another defined? I searched on the internet and in some books to no avail and it mostly appears in a special case - the ...
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1answer
543 views

Relative entropy between singular measures

Usually, to define relative entropy between two probability measures, one assumes absolute continuity. Is it possible to extend the usual definition in the non absolutely continuous case?
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Mutually singular measures with the same support

Let $X$ be a compact metric space and let $\mu$ be a measure on $(X,\mathcal{B})$, where $\mathcal{B}$ is the Borel $\sigma$-algebra of subsets of $X$. We define the support of $\mu$ as the smallest ...