Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [signed-measures]

The tag has no usage guidance.

0
votes
1answer
29 views

Finite integral of convolution

For appropriately well behaved functions $f(t) $, $g(t)$ the integral of their convolution is the product of their individual integrals. $$ \int_{-\infty}^{\infty}(f*g)(t) \, dt=\left(\int_{-\infty}^{...
0
votes
1answer
23 views

If $|\mu(E)|\le c\sqrt{\nu(E)}$ for all $E$ belonging to a generator, then the total variation of $\mu$ is bounded by $c\sqrt{\nu}$

Let $(\Omega,\mathcal A)$ be a measurable space, $\mu$ be a signed measure on $(\Omega,\mathcal A)$, $\nu$ be a measure on $(E,\mathcal E)$ and $c\ge0$ with $$\left|\mu(E)\right|\le c\sqrt{\nu(E)}\;\;\...
0
votes
0answers
21 views

Signed Measure equivalent definition

Let $\mu$ be a signed measure, define $$ |\mu|(E) = \mu^+(E) +\mu^-(E) $$ Why is it that $$ |{\mu}|(E) = \text{sup}\sum_{k=1}^n|\mu(E_k)|, $$ where the supremum is taken over all finite disjoint ...
0
votes
0answers
14 views

Can anyone prove this proposition? (Frame Functions)

Can anyone prove this proposition? It is proposition 3.2.2. out of A. Dvurecenskij's book on the applications of Gleason's Theorem. The proof in the book just says that it is obvious and doesn't ...
2
votes
0answers
29 views

Prove that $m$ is a positive measure.

Let $X$ be a compact metrizable space, and $m$ be a finite signed, Borel, regular measure on $X$. Assume that for every continuous function $f:X\to [0,\infty)$ one has $\int_Xfdm\ge0$. How to show $m$ ...
1
vote
0answers
28 views

Show that any nondecreasing $f:[0,\infty)\to\mathbb R$ of locally bounded variation admits a unique signed measure $\mu$ with $\mu([a,b])=f(b)-f(a)$

Let $f:[0,\infty)\to\mathbb R$ be right-continuous and $$v(t):=\operatorname{Var}_{[0,\:t]}f$$ denote the variation of $f$ on $[0,t]$ for $t\ge0$. Assume $v(t)<\infty$ for all $t\ge0$ and let $$f^\...
0
votes
1answer
20 views

Corollary from Hahn decomposition

let $v$ be a signed measure on $(X,\Sigma)$. let $A\in \Sigma$ such that $v(A)>0$. show that there is a $v$-positive set $B\subset A$ such that $v(A)\le v(B)$. I believe this statement can be ...
0
votes
1answer
27 views

Can the total variation measure of a complex measure be related to its real and imaginary parts?

If $\mu$ is a signed measure with Jordan decomposition $\mu=\mu^+-\mu^-$, then the total variation measure of $\mu$ is equal to $\mu^++\mu^-$. My question is, is it similarly possible to express the ...
2
votes
0answers
16 views

Characterization of Reflexive Subspaces of finite signed-measures

I'm looking for a characterization of the reflexive subspaces of the space of finite measures on a measurable space $(\Omega,\mathcal{F})$. Maybe something, more informative than the unit ball is ...
1
vote
1answer
36 views

An application of Lebesgue Differentiation Theorem

Let $\delta_0$ be the measure defined as $\delta_0(A)=1$ if $0 \in A$ and $\delta_0(A)=0$. Let $g_h(x)=\frac{1}{h}1_{[-h,h]}$ and define $\nu_h$ as: $\nu_h(A)=\int_{A}g_h(x)dx$. Show that for every ...
2
votes
1answer
47 views

Absolute continuity of measure and $\epsilon-\delta$ condtion

I am reading Folland's Real Analysis p. 89 Theorem: Let $\nu$ be a finite signed measure and $\mu$ a positive measure on $(X,M) $. Then $\nu << \mu$ iff ...
0
votes
2answers
128 views

Radon-Nikodym Derivative of a Total Variation Measure

Let $\nu$ be a signed measure which is absolutely continuous to a sigma-finite measure $\mu$. Show that $\frac{d|\nu|}{d\mu}=|\frac{d\nu}{d\mu}|$, where $|\nu|$ is the total variation measure of $\nu$...
1
vote
1answer
53 views

Equality of finite signed measures by showing that the integrals of every bounded continuous function coincide

In order to show the uniqueness of the Fourier coefficients of a signed measure, I need to show that : For any two finite signed measures $\nu_1, \nu_2$ on $\left([-\pi, \pi], \mathcal{B}_{[-\pi, ...
1
vote
1answer
105 views

Continuity of Lebesgue Stieltjes integral

I am trying to prove that Lebesgue-Stieltjes integral defines a cadlag function (i.e. right continuous with left limits) when its integrator is a cadlag function. Assume that $A(s)$, $s\in \mathbb{R}...
2
votes
1answer
82 views

An expression for a Total Variation Measure of a Signed Measure

Let $F$ be a Sigma algebra on a set $X$, and let $\mu$ and $\nu$ be probability measures on $(X,F)$, i.e. $\mu(X)=\nu(X)=1$. Finally let $\eta=\mu-\nu$. Show that $$|\eta|(X)=2\sup_{E\in F}|\eta(E)|,...
1
vote
1answer
69 views

Finding a Hahn Decomposition involving a Dirac Measure

Let $(X,F,\mu)$ be a finite measure space, i.e. $\mu(X)<\infty$. And let $x\in X$, and let $\delta_x$ be the Dirac measure with respect to $x$, i.e. $\delta_x(E)=1$ if $x\in E$ and $\delta_x(E)=0$ ...
0
votes
1answer
14 views

Prove that $\mu \cup \nu $ is the largest signed measure that is less than or equal to both $\mu,\nu$.

I am learning Signed Measures as a Part of my PhD coursework and I have got this exercise after just learning 1 page. I dont know how to proceed. Let $\mu ,\nu $ ne two finite measures on the ...
3
votes
1answer
32 views

Finding possible p's for signed measure

Im new here at StackExchange and hope some of you can help me with a problem I'm dealing with. We are considering a signed measure $\nu$ on $(\mathbb{N},2^{\mathbb{N}})$, which is given by $\nu(\{k\}...
3
votes
2answers
77 views

Signed measure of uncountable set

I have a question and hope some of you can help me :) Consider a signed measure $\nu$ on $(\Omega, \bf{A})$ and let be $P_i \in \bf{A}$ positive sets, such that $ \forall B \subset P_i: \nu(B) \geq0 $...
3
votes
1answer
101 views

Countably additive finite signed measures form a Banach Space.

I'm currently studying some topics in measure theory and I am not sure how to prove the following: Let $X$ a set, $\mathcal A$ a $\sigma$-algebra on X. Consider the set: $$ca(\mathcal A) = \{\mu:\...
2
votes
0answers
53 views

Correspondence between dual of the space of continuous maps and signed measures

On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found: Let $(X, d)$ be a compact metric. Recall that the dual ...
1
vote
1answer
30 views

Equivalence of the Variation of F$_\mu$(x) and $| \mu |$(($-\infty, x])$

Let $\mu$ be a finite signed Borel measure. I want to see why V$_{F_{\mu}}$($-\infty, x]$ = $| \mu | $ $((-\infty , x])$ for all x in R, where F$_\mu$(x) = $\mu((- \infty, x])$ and V$_{F_{\mu}}$($-\...
3
votes
0answers
32 views

If $\mathcal A$ is an algebra and $\mu$ is a vector measure on $\sigma(\mathcal A)$, can we approximate $\sigma(\mathcal A)$ by $\mathcal A$?

Let $\Omega$ be a set $\mathcal A$ be an algebra on $\Omega$ $E$ be a $\mathbb R$-vector space $\mu:\sigma(\mathcal A)\to E$ be $\sigma$-additive I want to show that for all $A\in\sigma(\mathcal A)...
1
vote
0answers
25 views

Characterizing the dual space of the linear space of the signed measures generated by a given set of measures.

Let $(\Omega,\mathcal{F})$ be a measurable space, $\varphi,\psi_{1},\ldots,\psi_{p}$ be real valued measurable functions on $(\Omega,\mathcal{F})$ and the set $\mathcal{M}$ of all non-negative ...
2
votes
1answer
79 views

If $\mu$ is a finite measure and $ν$ is a signed measure with $|\nu|\le C\mu$, are we able to show $\left|\frac{{\rm d}ν}{{\rm d}\mu}\right|\le C$?

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space $\nu$ be a signed measure on $(\Omega,\mathcal A)$ with $$|\nu(A)|\le C\mu(A)\;\;\;\text{for all }A\in\mathcal A\tag1$$ for some $C\ge0$ Note ...
2
votes
2answers
88 views

If $\mu$ is a finite measure and $\nu$ is a signed measure, can we extend the inequality $|\nu|\le C\mu$ from a generator to the whole σ-algebra?

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space $\mathcal E$ be a semiring with $\sigma(\mathcal E)=\mathcal A$ $\nu$ be a signed measure on $(\Omega,\mathcal A)$ with $$|\nu(A)|\le C\mu(A)\;\...
3
votes
0answers
43 views

Convergence of supremum of the series of a signed measure

Let $(X, \Sigma, \mu)$ be a measure space of finite measure and $f\in L^1 (\mu)$. For every $E\in \Sigma$ we define: $$v(E) = \int _ E f d\mu$$ Suppose that $f\in L^p (\mu)$ for some $p \in [1, + \...
1
vote
1answer
40 views

If $μ$ is a signed measure, $Ω^±$ is a Hahn decomposition and $μ^±$ is the Jordan decomposition, then $\int_{A∩Ω^±}X{\rm d}μ=\pm\int_AX{\rm d}μ^±$

Let $(\Omega,\mathcal A,\mu)$ be a finite signed measure space $(\Omega^+,\Omega^-)$ be a Hahn decomposition of $\Omega$ with respect to $\mu$ and $(\mu^+,\mu^-)$ denot the Jordan decomposition of $\...
2
votes
0answers
66 views

Show that $L^1(\nu)=L^1(|\nu|)$, where $\nu$ is a complex

Suppose that $\nu$ is a complex measure. Show that $L^1(\nu)=L^1(|\nu|)$. This is my attempt to prove this assertion. To begin with, suppose that $\mu=|\nu_r|+|\nu_i|$. Then $\nu_r <<\mu$ and $...