# 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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### 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 ...
1 vote
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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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Let $\mu$ be the Lebesgue-Stieltjes measure on $\mathbb{R}$ corresponding to the distribution function, $F$ where $$F(x) = \left\lbrace \begin{array}{ll} 0& \text{if} \,\, x<0\\ ... 0 votes 2 answers 197 views ### two mutually singular measures witch both support is whole \mathbf{R} I have some problem with an exercise(for homework): Find two mutually singular measures u and v (Borel finite on \mathbf{R}) with$$\mathrm{supp}(u)=\mathrm{supp}(v)=\mathbf{R}.$$I tried to ... 1 vote 0 answers 283 views ### 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 ... 1 vote 2 answers 287 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. 1 vote 2 answers 651 views ### 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 \... 1 vote 0 answers 111 views ### 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 ... 4 votes 2 answers 1k 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 ... 0 votes 1 answer 184 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 ... 6 votes 2 answers 2k 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 ... 2 votes 1 answer 3k 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\... 12 votes 3 answers 5k views ### 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 ... 1 vote 1 answer 318 views ### 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 ... 1 vote 0 answers 166 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\... 4 votes 2 answers 922 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. 2 votes 0 answers 346 views ### 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, ... 2 votes 0 answers 67 views ### 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... 0 votes 1 answer 1k 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 ... 0 votes 1 answer 739 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 ... 1 vote 1 answer 196 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} \...
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 ...