Skip to main content

Questions tagged [signed-measures]

A signed measure is a countably additive set function on a sigma-algebra and taking values in the extended reals, but not permitted to assign negative infinity to a set.

Filter by
Sorted by
Tagged with
0 votes
0 answers
19 views

Relation between total variation measure and total variation of a function

It is well known that if $f$ defined on a compact interval $[0,T]$ is a continuous function with finite variation, then $f$ induces a signed measure $\mu$ on $[0,T]$. Let $|\mu|$ be the total ...
George's user avatar
  • 75
0 votes
1 answer
48 views

Sum of two signed measures is well defined iff both do not take the same sign of $\pm \infty$

Let $(X,\mathcal{A})$ be a measurable space and $\mu$ and $\nu$ be a signed measures on $(X,\mathcal{A})$. We say that $\lambda =\mu+\nu$ is well-defined as a function $\lambda \: \mathcal{A} \to \...
Squirrel-Power's user avatar
0 votes
0 answers
51 views

Uniqueness of extension of signed measures

By Caratheordory's extension theorem, if the measure space is $\sigma-$ finite and we have two measures that are equal on an algebra generating the $\sigma-$ algbera, then they are equal. Will the ...
vkk's user avatar
  • 31
0 votes
0 answers
27 views

Sigma additivity of signed measure

Suppose we have $(X, M, v)$ as a measure space. $v$ is a signed measure, then it also satisfies sigma additivity, i.e. If {$E_j$} is a sequence of disjoint sets in $M$ , then $v(\cup_1^\infty E_j)=\...
Andrew_Ren's user avatar
5 votes
2 answers
155 views

Why are complex measures not allowed to attain $\infty$ while signed measures are?

I saw another question similar to this one but I'm not satisfied by answers. Here I changed the question to clearify the point I am interested in. I study Measure Theory for Real & Complex ...
Alileo's user avatar
  • 121
1 vote
1 answer
84 views

Characterization of absolute continuity for finite additive functions

Let $\nu:\mathbb{X}\longrightarrow \mathbb{R}$ and additive function, where $(X,\mathbb{X},\mu)$ is a measure space. That is, $\nu(\emptyset)=0$, and for any finite disjoint family of measurable sets $...
isaac098's user avatar
0 votes
1 answer
47 views

Clarification on the definition of signed measure: Is the rearrangement issue assumed to be impossible by the definition?

Let $(\Omega,M,\mu)$ be a $\underline{\text{signed}}$ measure space. Is it true that the definition of signed measure implicitly implies that Either for any sequence of disjoint measurable sets with ...
Asigan's user avatar
  • 1,759
1 vote
2 answers
102 views

Folland Theorem 3.22 Proof Explanation

At the first step of Theorem 3.22 of Folland, he mentioned that $d\nu = d\lambda + f\,dm$ implies $d|\nu| = d|\lambda| + |f| dm$: I am aware of the similar questions has been asked before on this ...
Partial T's user avatar
  • 561
0 votes
0 answers
51 views

Won't signed measures upset Riemann?

I can't seem to wrap my head around signed measures. If $\mu$ is a signed measure on a measurable space $(X, \Sigma)$, then there'll be sets of both positive and negative measure. Let $E_1, E_2, \...
Atom's user avatar
  • 3,965
1 vote
0 answers
35 views

boundedness of signed measures

Let us consider signed charges, these are finitely additive signed measures (I suppose this would also work with sigma additive signed measures). We work on a measure space $(\Omega, \mathcal{A})$, ...
guest1's user avatar
  • 351
1 vote
0 answers
30 views

(Signed) measures defined on algebra instead of sigma algebra

So some authors consider (sigma additive) signed measures on algebras (that are closed only under finite unions) instead of on sigma algebras. This happens for instance when considering the subset of ...
guest1's user avatar
  • 351
-1 votes
1 answer
56 views

Looking for books covering signed measures in detail

so I am looking for good text books that treat signed measures, integration thereof and total variation etc. In particular also for finitely additive signed measures. I have found Dunford and Schwartz ...
guest1's user avatar
  • 351
1 vote
1 answer
189 views

Space of finite signed measures

Given a measure space $(\Omega, \mathcal{A})$, let $\nu$ be a signed measure on that space and let $|\nu| := \nu^+ + \nu^-$ be the variation. Now consider the measurable functions $X:\Omega \...
guest1's user avatar
  • 351
3 votes
1 answer
164 views

Integration with respect to total variation norm measure

Let $\mu$ be a signed measure. Then it has the Jordan decomposition $\mu=\mu^{+}-\mu{-}$. The total variation is defined as $|\mu|=\mu^{+}+\mu{-}$. I am now wondering how to compute an integral $\int ...
guest1's user avatar
  • 351
0 votes
1 answer
43 views

Is this functional jointly continuous in the measure and the integrand?

Let $\Delta[0,1]$ be the space of all signed Borel measures over $[0,1]$, ranging between -1 and 1. $[0,1]^{[0,1]}$ is the set of all functions with domain $[0,1]$ that takes values in $[0,1]$. ...
Canine360's user avatar
  • 1,481
1 vote
0 answers
43 views

Cauchy-Schwarz-like inequality for a discrete quasi-probability distribution

Let $\mathbf{x},\mathbf{y},\mathbf{z}$ be pairwise orthonormal vectors from $\mathbb{R}^n$ where $n \geq 3$. For any $1 \leq i,j,k \leq n$ define $$p_{ijk} = \left| \begin{array}{ccc} x_i^2 & ...
meler's user avatar
  • 175
1 vote
0 answers
130 views

Folland lemma 3.7

I am struggling to understand the following theorem's proof on Folland's Real Analysis, page 89, lemma 3.7. Suppose that $\nu$ and $\mu$ are finite measures on $(X, \mathcal M)$. Either $\nu \perp \...
Squirrel-Power's user avatar
2 votes
1 answer
279 views

Why the difference of two measures is a signed measure?

Let $\mu_1: \mathcal{M} \to [0,\infty)$ be a finite measure and $\mu_2: \mathcal{M} \to [0,\infty]$ be a measure (can take infinite value). Now define $\mu: \mathcal{M} \to [-\infty,\infty]$ by $\mu = ...
FactorY's user avatar
  • 774
1 vote
0 answers
135 views

Triangle inequality for signed measures $| \mu +\nu |(A) \leq |\mu|(A) +|\nu|(A)$

I want to show that for finite signed measures $\mu, \nu$ defined on $\sigma$-algebra $\mathcal{A}$ the following triangle inequality holds $ | \mu +\nu |(A) \leq |\mu|(A) +|\nu|(A) $ for all $A\in \...
TOMILO87's user avatar
  • 510
4 votes
0 answers
76 views

Outer signed measure

I would like to ask whether there is some kind of analogue of outer measure when dealing with signed measures. I would like to assign measure to all the subsets, not just some $\sigma$-field. I'm ...
Jkbb's user avatar
  • 492
1 vote
0 answers
102 views

If integral of every continuous function is zero then the measure is zero

Here's what I am trying to prove: Let $X$ be a topological space and $\mu$ be a finite signed Borel measure on $X$. Suppose that \begin{align*} \int f d\mu =0 \end{align*} for every continuous ...
ashK's user avatar
  • 4,015
0 votes
0 answers
37 views

Equivalent formulation of the variation of a signed measure

I am going through Bogachev's book on Measure Theory, and having read the definition of variation of measures I am trying to grasp what it is doing. I have the following doubt. Let $\nu$ be a signed ...
F f F's user avatar
  • 65
1 vote
1 answer
129 views

Need help understanding definition of total variation metric of two probability measures

Fix a measurable space $(X,\Sigma)$. Let $\mu:\Sigma\to\mathbb{R}$ be some signed measure. Then the total variation norm is defined by $$ \|\mu\|_{\textrm{TV}}=\mu_{+}(X)+\mu_{-}(X),\quad (*) $$ where ...
Miski123's user avatar
  • 307
0 votes
0 answers
58 views

Question regarding derivative and measures

Here's what I need to show: Let $\mu$ be any finite signed Borel measure on $[-\pi,\pi]$. Let $g: [-\pi,\pi] \to \mathbb R$ be defined by $g(\theta ) = \int_{0}^{\theta} d\mu (t)$ if $\theta >0$ ...
ashK's user avatar
  • 4,015
2 votes
1 answer
91 views

A couple of questions regarding a short proof of the Jordan Decomposition Theorem.

Theorem 6.21 (Jordan decomposition). If $\nu$ is a signed measure on a measurable space $(X, \mathcal{A})$, then there exist unique measures $\nu^+, \nu^- \colon \mathcal{A} \to [0, \infty]$, one of ...
Sam's user avatar
  • 4,900
1 vote
1 answer
71 views

Let $\mu,\nu$ be two finite positive measures. Show that, there is a set $E$ such that $\mu_E\ll\nu_E$ and $\mu_{E^c}\perp\nu_{E^c}$

Let $\mu,\nu$ be two finite positive measures on $(X,\mathfrak{M})$. For $E\in\mathfrak{M}$, we define $\mu_E(A):=\mu(A\cap E)$ and similarly $\nu_E$. Show that, there is $E\in\mathfrak{M}$ such that $...
DeltaEpsilon's user avatar
  • 1,120
1 vote
0 answers
46 views

A Question about Real Analysis by Folland

The problem is that Suppose $\nu(E)=\int fd\mu$ where $\mu$ is a positive measure and $f$ is an extended $\mu$-integrable function. Describe the Hahn decompositions of $\nu$ and the positive, ...
Ethan's user avatar
  • 21
0 votes
3 answers
86 views

How to prove $\nu(A) := \sum_{k\in A}a_k$ is a signed measure on $(\mathbb{N},\mathcal{P}(\mathbb{N}))$

I'm currently reading about signed measures. In doing so, we early on give an example of a signed measure (before the Hanh or Lebesgue-Radon-Nikodym Decomposition theorems). In particular, we let let $...
Simon SMN's user avatar
  • 173
3 votes
2 answers
179 views

Definition of a signed measure in Folland and its meaning when the signed measure is not finite

So I'm currently reading about signed measures in "Real analysis" by Folland. In it, he defines a signed measure as follows. Let $(X,\mathcal{M})$ be a measurable space. Then a signed ...
Simon SMN's user avatar
  • 173
3 votes
0 answers
90 views

Portmanteau theorem for finite signed Borel measures

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$, and $\mathcal C_b(X)$ be ...
Analyst's user avatar
  • 5,667
3 votes
0 answers
81 views

Let $\mu_n,\mu \in \mathcal M(X)$ such that $\mu_n \rightharpoonup \mu$ and $[\mu_n] \to [\mu]$, then $|\mu_n| \rightharpoonup |\mu|$

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$, and $\mathcal C_b(X)$ be ...
Analyst's user avatar
  • 5,667
2 votes
1 answer
73 views

Does $\mu_n \overset{1}{\rightharpoonup} \mu$ necessarily imply $\mu_n \overset{2}{\rightharpoonup} \mu$?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$, and $\mathcal C_0(X)...
Akira's user avatar
  • 17.5k
3 votes
1 answer
120 views

Does $\mu_n \overset{*}{\rightharpoonup} \mu$ (or $\mu_n \rightharpoonup \mu$) imply $\{[\mu_n] \mid n \in \mathbb N\}$ is bounded?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$, and $\mathcal C_0(X)...
Akira's user avatar
  • 17.5k
4 votes
2 answers
351 views

How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous functions, $\mathcal C_0(X)$ be the ...
Analyst's user avatar
  • 5,667
2 votes
1 answer
96 views

Representation of Functional in the Dual of the Space of Signed Measures on a Measurable Space

I suspect the following is true, but I don't know a reference. Let $X$ be a measurable space (with $\sigma$-algebra $\mathcal{F}(X)$), and let $MX$ denote the space of finite signed measures on $X$. ...
HardyHulley's user avatar
0 votes
0 answers
55 views

Why signed measures should respect countable additivity and not sub-addivity.

$\newcommand{\scrF}{\mathcal{F}}$ Some definitions first. Defn: Let $(\Omega, \scrF)$ be a measurable space. A signed measure on $(\Omega, \scrF)$ is a set-theoretic function $\mu:\scrF \rightarrow (-...
Irving Rabin's user avatar
  • 2,663
3 votes
0 answers
394 views

$3$ versions of Riesz–Markov–Kakutani theorem

I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek. I have no question here. This thread is to summarize $3$ versions of the theorem (in an increasing order of generality). I ...
Analyst's user avatar
  • 5,667
3 votes
1 answer
300 views

What does it mean for a signed measure to be "regular" in Riesz-Markov-Kakutani theorem?

I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek. Theorem 3.23 (Riesz-Markov-Kakutani for $\left.C_{0}(\boldsymbol{X})^{*}\right)$. Let $X$ be a locally compact Hausdorff ...
Analyst's user avatar
  • 5,667
4 votes
1 answer
104 views

If $\mu_n \overset{\ast}{\rightharpoonup}\mu$, then $\mu^+_n \overset{\ast}{\rightharpoonup} \mu^+$ and $\mu^-_n \overset{\ast}{\rightharpoonup}\mu^-$

Let $X$ be a Polish space and $\mu, \mu_n$ finite signed Borel measures on $X$. Assume that $\mu_n \overset{\ast}{\rightharpoonup} \mu$, i.e., $$ \int_X f \mathrm d \mu_n \to \int_X f \mathrm d \mu $$ ...
Akira's user avatar
  • 17.5k
0 votes
1 answer
353 views

How to find total variation of a signed measure??

Let $(X,F,\nu)$ be a signed measure on the sigma algebra F. now by Jordan-Hahn decomposition theorem $\nu = \nu_1 - \nu_2$, where $\nu_1$ and $\nu_2$ are positive and mutually singular, and such ...
Vishnudasa Srinivasan's user avatar
1 vote
0 answers
24 views

Let $L = \{f:X \to [0, 1] \mid f \text{ is measurable}\}$. Then $\mu_+(X) = \sup_{f \in L} \int_X f \mathrm d \mu$

I'm trying to prove this characterization. Could you have a check if my attempt is fine? Let $X$ be a topological space and $\mu$ a signed Borel measure on $X$. Let $\mu = \mu_+ - \mu_-$ be the ...
Akira's user avatar
  • 17.5k
1 vote
0 answers
54 views

What is the equivalent operation of measures to the multiplication of (right-continuous, bounded variation) function

Consider the spaces $\mathcal{M}$ the set of all finite, signed measures on $(\mathbb{R},\mathcal{B})$ $\Phi$ the set of all right-continuous functions $F:\mathbb{R}\to\mathbb{R}$ with bounded ...
tadeseus's user avatar
0 votes
1 answer
118 views

If $\lambda = \mu-\nu$, is it true that $\int f \mathrm d \lambda^+- \int f \mathrm d \lambda^- = \int f \mathrm d \mu -\int f \mathrm d \nu$?

People usually write $\int f \mathrm d (\mu-\nu)$ as a shorthand for $\int f \mathrm d \mu -\int f \mathrm d \nu$. Let $(\Omega, \mathcal F)$ be a measure space and $\mathcal M :=\mathcal M(\Omega)$ ...
Akira's user avatar
  • 17.5k
0 votes
0 answers
34 views

Textbooks about integration w.r.t. signed measures

I'm reading this question about signed measure. Assume that $\lambda$ is a finite signed measure and have a Jordan decomposition $\lambda = \lambda^+ - \lambda^-$. Then $$ \int f \mathrm d \lambda := ...
Akira's user avatar
  • 17.5k
2 votes
1 answer
102 views

Show that a set function is sigma additive on an algebra, but not extendable to a signed measure on the generated sigma algebra.

I am currently preparing for a measure theory exam and struggling with the following problem: Consider the algebra \begin{align} \mathfrak A = \{ A \subseteq \mathbb R: |A| < \infty ~ \text{or} ...
Richard Weiss's user avatar
1 vote
0 answers
58 views

For which $p \in \mathbb R$ does a signed measure on $([0, 1], \mathcal B([0, 1]))$ exist such that $\mu([0, x]) = x^p \sin(1/x)$?

I am currently preparing for a measure theory exam and struggling with the following problem: For which $p \in \mathbb R$ does a signed measure on $([0, 1], \mathcal B([0, 1]))$ exist such that $\mu([...
Richard Weiss's user avatar
1 vote
2 answers
64 views

On the null set of absolute continuous signed measure

Folland states the absolute continuity of a signed measure as follows: If $\nu \ll \mu$, then $ \mu(E)=0$ implies that $E$ is a null set for $\nu$ because $\forall A \subset E : \mu(A)=0$ and so $\nu(...
khashayar's user avatar
  • 2,214
1 vote
1 answer
119 views

Finiteness of signed measure

Let $\nu$ be a signed measure and $|\nu|$ be its total variation. Then does $\nu \text{ finite } \implies |\nu| \text{ finite} $? If so, why? I can almost see this by appealing to the Jordan ...
ashman's user avatar
  • 962
4 votes
0 answers
214 views

Is it possible to determine the sign of the determinant of a matrix without knowing/using the formula for the determinant?

I'm trying to build intuition for the orientation of a set of vectors independent of the well-known definition of the determinant. My intuition wants to go something like this: any set of vectors can ...
Jagerber48's user avatar
  • 1,441
2 votes
0 answers
52 views

When are signed measures isomorphic to a RKHS?

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions over $X$, with a bounded kernel $\mathcal{K}: X\times X\to \mathbb{R}$. Let us assume there is a sigma-algebra over $X$ such that $\...
sadkangaroo's user avatar