# 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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### 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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### 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 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 ...
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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### 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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### 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 ...
1answer
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### 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 ...