Questions tagged [radon-nikodym]

For questions involving the notion of the Radon-Nikodym derivative or the Radon-Nikodym theorem. Use this tag along with (probability-theory) or (measure-theory).

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2
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1answer
35 views

Can we show $\frac{\nu\left(B_\varepsilon(x)\right)}{\mu\left(B_\varepsilon(x)\right)}\xrightarrow{\varepsilon\to0+}\frac{{\rm d}\nu}{{\rm d}\mu}(x)$?

Let $(E,\mathcal E,\mu)$ be a finite measure space and $\nu$ be a finite signed measure on $(E,\mathcal E)$ with $\nu\ll\mu$. By the Radon-Nikodým theorem, $$\nu=f\mu\tag1$$ for some $f\in L^1(\mu)$. ...
3
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0answers
13 views

Extending proof of positive measure to real measure for Radon Nikodym

The Wikipedia proof of Radon-Nikodym first shows for positive measures, then extends to real measures by applying the positive measure Radon-Nikodym theorem to the unique + and - (positive) measures ...
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0answers
36 views

Constructing a specific measure from a given measure such that its Rdon Nikodym derivative takes specific values

Take a measure space $(\Omega, \mathcal{A}, \tau, P)$, with $P$ being a probability measure, and $\tau$ being a topology on $\Omega$. I would like to construct a measure $Q$ such that $Q$ is ...
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26 views

Integrating a measure with respect to itself

Take a measure space $(\Omega, \mathcal{A}, P)$, with $P$ being a probability measure. I want to construct a second measure from $P$ via $Q(E) := \int_E P(\omega) \; dP(\omega)$. In other words, I ...
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0answers
20 views

Finding Radon Nikodym derivative for a function of the measure

I have a question regarding finding the Radon Nikodym derivative of a measure w.r.t. another measure constructed from the first one. The situation is the following: Take a measure space $(\Omega, \...
0
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0answers
14 views

How to show the following equality (relating to measure theory)?

Let $(\mathcal{X},\mathcal{A})$ be some measurable space, and let $P$, $Q$ and $\nu$ be probability measures defined on this space, with $\nu = \tfrac{1}{2}(P+Q)$. From this I know that $\nu$ ...
0
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1answer
32 views

Exercise on Radon–Nikodym derivative

Consider the measure space $(\mathbb{N}, P(\mathbb{N}))$ and two measures $\mu$ and $\nu$, where $\mu$ is the counting measure and $\nu$ is defined as $$ \nu(E) = \sum_{n \in E}(n+1)$$ Compute $d\mu/...
0
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1answer
28 views

Clarifying expectation with respect to a function, and the Radon-Nikodym derivative

I am trying to understand induced measures in the context of expectation. I have commonly seen formulas such as KL-divergence written like so: $$\text{KL}(p||q) = \int\log\dfrac{p(x)}{q(x)}dp = \int p(...
1
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1answer
25 views

Necessity of probability in $P \ll \mu \iff \forall \varepsilon > 0 \exists \delta > 0 : \forall A, \mu(A) < \delta , P(A) \leq \epsilon$

Given the following proposition : Let $P$ be a probability and $\mu$ a $\sigma$ finite measure. Then $P \ll \mu \iff \forall \hspace{0.1cm} \varepsilon > 0 \hspace{0.1cm} \exists\hspace{0.1cm} \...
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0answers
19 views

Variation of the Radon-Nikodym Theorem

If I have two finite measures $\nu$ and $\mu$ such that $\nu$ is unsigned and $\mu$ is signed, and $\nu \ll \mu$, is it possible to extend the Radon-Nikodym Theorem to these to find $\frac{d\nu}{d\mu}$...
4
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1answer
65 views

Generalization of Radon-Nikodym Theorem

I'm wondering if the following statement holds: Let $(X, \mathfrak{B})$ be a measurable space, and $\mu, \nu$ be complex measures. Assume $\nu << \mu$. Is there a complex valued measurable ...
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0answers
16 views

Joint measure and absolute continuity wrt product of marginals

I am trying to build intuition over the following matter: let $X,Y$ be two random variables with corresponding probability measures $P_X,P_Y$. Assume also there exists a joint measure $P_{XY}$ such ...
6
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303 views

Has Math of Finance education become unnecessarily inaccessible?

EDIT: Post shortened, following the suggestions of @Noah Schweber (Thank you!) EDIT#2: I have spent several hours trying to properly form these questions, so instead of voting this down, please ...
2
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1answer
45 views

Is Radon-Nikodym derivative with respect to a finite measure real-valued a.e.?

This question comes from this question. The answer therein missed an argument that the Radon-Nikodym derivative is real-valued a.e. Without this, the proof in that answer has flaw because either the ...
1
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0answers
18 views

Prove $\lim_{\epsilon\to0^+}\frac{1}{\epsilon}\int_{(t,t+\epsilon)}pd\lambda=p(t)$, $p$ the Radon-Nikodym derivative

If $\mu$ is a measure on $(\mathbb{R}_+, \mathcal{B}(\mathbb{R}_+))$ such that $\mu([0,t])$ is finite for all $t\in\mathbb{R}_+$, where $\mathbb{R}_+=[0,+\infty)$ and $\mathcal{B}(\mathbb{R}_+)$ is ...
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0answers
33 views

Bounded Radon-Nikodym derivative of a measure

Let $Y\subset\mathbb R^d$ and $X\subset\mathbb R$. Let $\mu\in\mathcal P(Y)$ be absolutely continuous with a density still denoted $\mu$ which is bounded with compact support (so $\mu(E)\leq\|\mu\|_{\...
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1answer
15 views

Expression of density function f, through derivatives of measures.

Lemma: Let $P$ and $\mu$ be probability measures on a measurable space $(\Omega, \mathcal{A})$ with restrictions $P_{m}$ and $\mu_{m}$ to the elements of an increasing sequence of σ-fields $\mathcal{...
4
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1answer
73 views

Decomposition of the variation of a signed measure as $|\mu|(A) = \int_A |\frac{d\mu_{1a}}{d\mu_2}-1|d\mu_2 + \mu_{1s}(A)$, where $\mu=\mu_1-\mu_2$

Let $\mu_1$ and $\mu_2$ be two finite measures on $(\Omega, \mathcal{F})$. Let $\mu_1 = \mu_{1a}+\mu_{1s}$ be the Lebesgue decomposition of $\mu_1$ w.r.t. $\mu_2$, that is, $\mu_{1a} \ll \mu_2$ and $\...
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0answers
39 views

Are there “continuous” random variables without a density function?

Assume you have a random varaible with a cumulation distribution function $F(x)=P(X\le x)$. This function will always be right-continuous, but assume that it also is left-continuous. Then there are no ...
2
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1answer
47 views

A Radon-Nikodym-type theorem due to S. Sakai

I got stuck with the following problem while going through the proof of Lemma $2^{\circ}$, Chapter 10 from the book 'Lectures on von Neumann Algebras' by Strătilă and Zsidó. Problem: Let $\mathscr{M}$ ...
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1answer
40 views

Problems in understanding a step from a proof of the Radon-Nikodym theorem

From page 115 of "Measure theory and probability theory" by Krishna and Soumendra, as a step of the Radon-Nikodym theorem's proof: Suppose that $\mu_1$ and $\mu_2$ are finite measures. Let $...
2
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0answers
43 views

The GNS construction and a Radon-Nikodym-like formula

Let $\mathcal{A}$ be a $W^{*}$-algebra, $\omega$ a normal, positive linear functional on $\mathcal{A}$, $\mathbf{p}$ the support projection of $\omega$, and $(\mathcal{H},\pi,\psi)$ the GNS triple ...
4
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1answer
67 views

Does there exist an absolutely continuous probability measure on every measure space?

Let $(\Omega,\mathcal F,\mu)$ be an arbitrary measure space, where $\mu$ is non-zero but does not need to be $\sigma$-finite or semi-finite. Does there necessarily exist a probability measure $P$ on $(...
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0answers
22 views

Question about absolutely continuous distribution definition

So my book defines that a random variable $X$ has absolutely continuous distribution $F$, if it can be represented as $F(x)=\int_{-\infty}^x p(\mu)\,d\mu$ for some non-negative and integrable function ...
12
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1answer
260 views

Show that an increasing function has derivative $0$ a.e.

Let $0<p<1$ and define $F:[0,1]\rightarrow[0,1]$ by $$F(x)=\begin{cases} pF(2x),&x\in\left[0,\frac12\right]\\ p+qF(2x-1),&x\in\left[\frac12,1\right] \end{cases}$$ where $q=1-p$. I would ...
4
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1answer
89 views

Understanding the Lebesgue's Decomposition Theorem

In his book Bauer proves the Lebesgue's decomposition theorem. Actually he proves it only in the case where $\mu$ and $\nu$ are finite, leaving the $\sigma$-finite case as an exercise. Looking at the ...
0
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1answer
55 views

Signed Measure $\nu$ mapping $A \mapsto \int_A f d\mu$ and the Radon-Nikodym derivative

I am supposed to take an exam in August and so I am trying to prepare (this is not homework). So far I am pretty good at most topics, but anything related to Radon-Nikodym I don't quite understand. ...
4
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0answers
56 views

Radon-Nikodym Derivative: Proposition 3.9 Folland

Suppose that $\nu$ is a $\sigma$-finite signed measure and $\mu, \lambda$ are $\sigma$-finite positive measures on $(X, \mathcal{M})$ such that $\nu \ll \mu$ and $\mu \ll \lambda$. Then, the following ...
4
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0answers
57 views

Question on Radon-Nikodym derivatives

In Folland's Real Analysis, there is this theorem: Suppose that $\nu$ is a $\sigma$-finite signed measure and $\mu, \lambda$ are $\sigma$-finite positive measures on $(X, \mathcal{M})$ such that $\nu \...
3
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0answers
49 views

Help with finding the Lebesgue decomposition of measures

Consider the increasing, right-continuous function $$ F(x) = \begin{cases} 0 &x < 0 \\ 1+x &x \geq 0 \end{cases} $$ and let $\nu = \nu_F$ be the associated Borel ...
0
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1answer
79 views

Does a Lebesgue measurable set E exist with this property?

Does there exist a Lebesgue measurable set $E$ such that for all $n>0$ $m(E \bigcap [0,n])=\frac{n}{2}$? I think we can't have such this set, because if so, then we can write $E$ as a countable ...
0
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0answers
32 views

Integral Identity on Measure Theory [duplicate]

I'm trying to prove the following identity: If $f:X \rightarrow \mathbb{R}$ is a bounded, continuous function, and $\mu$ is a borel measure in $X$, then $\int f d\mu = \int_0^1 \mu(\{x \in X ; f(x)>...
2
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0answers
87 views

Weak convergence of Radon-Nikodým derivatives

I am working at the moment with a sequence of finite measures $\nu_n << \mu_n$ converging weakly to $\nu << \mu$. I have seen that if the Radon-Nikodým derivatives $h_n := \dfrac{d\nu_n}{d\...
3
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1answer
35 views

What is the RN derivative of infinite product measure?

Suppose $\mu_k$ and $\nu_k$, $k=1,2,...$ are sigma-finite measures on spaces $(S_k,\mathcal F_k)$ such that $\nu_k<<\mu_k$ for each $k$. Let $f_k=\dfrac{d\nu_k}{d\mu_k}$ for each $k$. Then is it ...
0
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2answers
71 views

Question regarding definition of $\mathbb{P}(A||\mathscr{G})$

I'm currently study conditional probaility form the book Probability and measure Book by Patrick Billingsley and I stuck to understanding the following thing. We know that $\mathbb{P}(A||\mathscr{G})$ ...
1
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1answer
28 views

Finding Radon-Nikodym derivative $d\mu/dm$ where $m$ is the Lebesgue measure on $[0,1]$, $f(x)=x^2$, and $\mu(E)=m(f(E))$

Consider the function $f:[0,1]\to \Bbb R$, $f(x)=x^2$. Let $m$ denote the Lebesgue measure on $[0,1]$ and define $\mu(E)=m(f(E))$. Since $f$ is absolutely continuous and nondecreasing, $f$ maps ...
0
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1answer
54 views

Radon-Nikodym Theorem for two positive measures

Let $\mu,\nu$ be two positive measures on $(X,\mathscr{A})$ and $\mu$ is finite. If $\nu \ll \mu$, then does there exist a measurable function $f: X \to [0,\infty]$ such that $$ \nu(E) = \int_E f d \...
0
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1answer
31 views

An application of Radon-Nicodym theorem

Consider $M$ be the $\sigma -$algebra of Lebesgue measurable sets and $\mu $ the Lebesgue measure. Denote by $P$ the set of $p-$measurable sets, that is the sets $A\in \mathcal{P}\left( \mathbb{R} \...
0
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0answers
34 views

Show that $\forall B\in\mathscr{F},\int_B\nu(dy)k(x,y)$ is $\mathscr{E}$-measurable given $\nu$ is $\sigma$-finite

This is a question from Erhan Cinlar's "Probability and Stochastics" Chapter 1 Section 6 Question 6.30: Let $\nu$ be a $\sigma$-finite measure on $(F,\mathscr{F})$, and let $k$ be a ...
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0answers
69 views

Radon-Nikodym Theorem (Folland 3.8)

I am reading the proof of Radon-Nikodym Theorem in Follands "Real Analysis Modern Techniques and Their Applications". Specifically, Theorem 3.8 on page 90. Folland starts with the case that both the ...
0
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2answers
88 views

absolutely continuous with respect to a finite measure, then $\Sigma$-finite

This is a question from Erhan Cinlar's Probability and Stochastics book: If $\mu$ is absolutely continuous with respect to a finite measure $\nu$, then $\mu$ is $\Sigma$-finite. Can i have a hint on ...
1
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1answer
38 views

Does joint distribution affect Radon-Nikodym derivative?

Given two real-valued random variables $X, Y$ with distributions $\mu_X, \mu_Y$. Suppose $\mu_X<\!<\mu_Y$, then the Radon-Nikodym derivative $\frac{d\mu_Y}{d\mu_X}(\cdot)$ exists $\mu_X$-a.e. on ...
1
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0answers
33 views

Understanding a lemma from Bauer's book

This lemma precedes the proof of the Radon-Nikodym Theorem in Bauer's book Measure and Integration Theory. At the beginning of the proof he writes "we may obviously suppose that $\varrho(\Omega)\geq0$...
1
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1answer
117 views

Radon-Nikodym Derivative of a Mixed Distribution

When we have a continuous distribution $F_X(x)$, we can take the Radon-Nikodym derivative (RND) of the probability measure with respect to Lebesgue measure to get the density $f_X(x)$. When we have ...
2
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0answers
43 views

Beyond Radon-Nykodim: Besicovitch theorem

I found, with no reference, the following theorem which is called Besicovitch derivation theorem. Do you know any article/book where I can find this powerful result? Let $\Omega\subset \mathbb R^n$ ...
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1answer
49 views

When is the Radon-Nikodym derivative (essentially) bounded?

Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{esssup}(\frac{d\mu}{d\nu})<\infty$? Hypothesis:...
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1answer
27 views

Show that $\frac{d(\alpha\times \beta)}{d( \mu\times \nu)}(x,y)=\frac{d\alpha}{d\mu}(x)\frac{d\beta}{d\nu}(y)$

Suppose $(X,\mathcal M,\alpha)$ and $(Y,\mathcal N,\beta)$ are positive finite measure spaces and $(X,\mathcal M,\mu)$ and $(Y,\mathcal N,\nu)$ are positive $\sigma$-finite measure spaces with $\alpha\...
0
votes
1answer
49 views

Radon Nikodym derivative of a restriction of a measure.

So how I see it $\nu <<\mu $ on $\mathcal{N}$. Thus there exists a Borel measurable function such that $\nu(A)=\int_Ahd\mu|\mathcal{N}=\int_Ahd\mu$ where the equality holds by the observations ...
1
vote
1answer
30 views

If $\lambda$ is a signed measure and $\lambda\ll\mu$, then $\lambda^{+}\ll\mu$ and $\lambda^{-}\ll\mu$

I was trying to prove the Radon-Nikodym theorem for complex measures, so I tried to decompose the complex measure first into its real and imaginary parts, which are signed measures, and then each of ...
1
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1answer
81 views

Reference Request: Finite Borel measure are Radon

Original: Let $\mu$ be a finite measure on $\mathbb{R}^k$ dominated by the Lebesgue measure. Does $\mu$ need to be a Radon measure? Updated Question: Does anyone have a reference to the fact that if ...