Questions tagged [signed-measures]

Question 1

For unsigned measure $\mu$ on a measurable space $(X, \mathcal B)$, we have: (Downwards monotone convergence) If ${E_1 \supset E_2 \supset \ldots}$ are ${{\mathcal B}}$-measurable, and ${\mu(E_n)<\...

Question 2

Given a finite signed measure $\mu$ on a $\sigma$-algebra $\mathcal{A}$, we say that $\mathcal{K}$ is an approximating class for $\mu$ on $\mathcal{A}$ if, for every $A\in\mathcal{A},\varepsilon>0$ ...

Question 3

I'm reading Measure Theory, Probability and Stochastic Processes by Le Gall. I'm stuck on a claim in a theorem (Theorem 6.2) on the total variation of a signed measure. Consider the definition of ...

Question 4

If $\mu$ is a bounded signed measure on a measurable space $(\mathcal{F}, \Omega)$, then there exist $A_1, A_2$ in the space satisfying the following: $$ \mu(A_1) = \max \{\mu(A)\}, \\ \mu(A_2) = \min ...

Question 5

Let $\mu$ be a signed measure on $X$ with Jordan decomposition $\mu^+-\mu^-$, $\pi:X\rightarrow Y$ a measurable map, $\pi_\#\mu$ the pushforward measure of $\mu$ with respect to $\pi$. If $f:Y\...

Question 6

Let $X$ be a Hausdorff space and $\mu,\nu:\mathfrak{B}(X)\to \mathbb{R}$ two Radon signed measure (that is, $|\mu|$ and $|\nu|$ are Radon measures). If $\mu (K)=\nu (K)$ for all compacts $K\subseteq X$...

Question 7

Let $(X,\mathcal{M},\mu)$ be a measure space and let $f:X \to [-\infty,\infty]$ be an extended $\mu$-integrable function (i.e. $f$ is measurable and at least one of $\int f^{+} \,d\mu$ and $\int f^{-} ...

Question 8

Let $(X,\mathcal{B})$ be a measurable space where $X \subseteq \mathbb{R}^d$,compact, $\mathcal{B}$ is the Borel sigma algebra, $\mu$ is a finite signed measure. Let $\left\langle \mu_n\right\rangle_{...

Question 9

I am interested in the properties of smooth quasi-probability density functions. In two dimensions, these are smooth functions $\rho(x,y):\, \mathbb{R}^2\to \mathbb{R}$ that satisfy the conditions \...

Question 10

This is a question from Richard Bass, Real Analysis for Graduate Students. If $\mu$ is a signed measure on $(X,A)$ and $|\mu|$ is the total variation measure, prove that there exists a real-valued ...

Question 11

I need to prove the following result: Remark 4.30$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Let $B(X,\mathscr{A},\mathbb{R})$ be ...

Question 12

Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable basis of the topology of $X$, which we can assume to be closed under finite unions and intersections, ...

Question 13

I am self-studying signed measure.I got stuck on the following remark: Remark$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Then the ...

Question 14

I am self-studying signed measure, and I come across the following construction: Let $\mu$ be a signed measure on the measurable space $(X,\mathscr{A})$, and let $A$ be a subset of $X$ that belongs ...

Question 15

I am self-studying measure theory and I come across the following question: Prove: Let $(X,\mathscr{A},\mu)$ be a measure space, let $f$ belong to $\mathscr{L}^1(X,\mathscr{A},\mu,\mathbb{R})$, and ...

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Questions tagged [signed-measures]

Downwards monotone convergence is unavailable for signed measures

Approximating class for signed measure

Total variation is a measure; Le Gall

About max and min of bounded signed measure

Is change of variables formula for signed measures valid also for functions with infinite integral?

Are two Radon signed measures agreeing on compacts equal?

Showing that $\nu(E) := \int_E f \,d\mu$ is a signed measure when $f$ is extended $\mu$-integrable

If $\mu_n {\rightharpoonup}\mu$, then under what conditions $|\mu_n - \mu | {\rightharpoonup} 0$?

Quasi-probability density functions and vanishing expectation

Signed Measures and Radon-Nikodym Theorem with Total Variation Measure

Prove: $F:M(X,\mathscr{A},\mathbb{R})\to\mathbb{R}$ defined by $F(\mu)=\int fd\mu$ is a linear functional. [closed]

Signed measure and bounded total variation on algebras

Why does $\mu\mapsto\int fd\mu$ defines a linear functional on $M(X,\mathscr{A},\mathbb{R})$?

Question About Signed Measures

Prove: Define $\nu$ on $\mathscr{A}$ by $\nu(A) = \int_Afd\mu$. Then $\nu$ is a signed measure on $(X,\mathscr{A})$.

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