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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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Absolute continuity of conditionals and marginals

I am beginning my study of measure theory and I came across the Radon-Nikodym Theorem (and derivative). I tried to play with the concept around with "elementary" objects but I am having troubles ...
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Conditional Expectation and Radon-Nikodyn

I have a question about the proof of the existence of a conditional expectation using the Radon-Nikodyn derivative. Let $X$ be a nonnegative random variable defined on $(\Omega, \mathcal{F}_0, \...
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Radon-Nikodym derivative after passing through Markov kernel

In the context of research in the field of $f$-divergences, I am using measure theory to prove some nice properties. The following result is really useful and so I want to be sure I understand it ...
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Extended metric on the space of probability measures

Let $P(X,\mathcal X)$ denote the collection of probability measures on some measurable space $(X,\mathcal X)$. For $\mu,\nu$ in this space, define $$ d(\mu,\nu)=\log \frac{\operatorname{ess\ sup}_{\...
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Replacing differentiation with anti-integration?

When thinking about the proof of the differentiability of Taylor series, I noticed that the theorem was proved by using properties of integrals. This got me thinking: To what extent can the role of ...
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Question about Radon measures in $\mathbb{R}^n$

Consider the following: Let $f$ be a $L^1(\mathbb{R}^n)$ function and define the Borel measure $d\mu=fdx$, where $dx$ is the Lebesgue measure. Now, since $f$ is in $L^1(\mathbb{R}^n)$, we know that $...
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Convergence of KL-divergence along a convergent sequence of measures

My question is about Lemma 12 and 13 (page 6) of of this paper https://arxiv.org/abs/1802.09583. The Lemma 13 in particular proves, ``Let $\log(g)$ be bounded. If $P_n \rightarrow P$, then $KL((P_n)_g ...
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Ito Diffusion with Change of Measure

Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
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If an integral is P-measurable with respect to a G measure, then this integral will be P-measurable with respect to a PN measure, since $PN \ll G$?

I am struggling to prove a result, which I don't know if it is possible as I have not found any reference. I will write some information that I don't know if they are really necessary to solve my ...
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Finding $f$ such that $Q(B)=\int_B f d\lambda$

Let $X:(\Omega,\mathscr{A},\mathbb{P})\to (\mathbb{R},\mathscr{B}_{\mathbb{R}})$. $A\in\mathscr{B}_{\mathbb{R}}$ such that $0<\lambda(A)<\infty$ and let $Q(B)=\frac{\lambda(A\cap B)}{\lambda(A)...
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Proof of an Abstract Bayes' Theorem

In Björk (2009) a Bayes' theorem is given by Assume that $X$ is a random variable on $(\Omega, \mathcal F, P)$ and let $Q$ be another probability measure on $(\Omega, \mathcal F)$ with Radon-...
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Absolutely continuous measure and equality to $0$

In the context of proving that the data processing inequality for $f$-divergences hold for any Markov kernel I am interested in the following statement If $\mu$ and $\nu$ are two probability ...
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Density of “opposite” measure

Let $\mathbb{P}$ be a complete probability measure on the measurable space $(\Omega,\mathcal{F})$. Define the measure $\mathbb{Q}$ on measurable sets $A \in \mathcal{F}$ by $$ \mathbb{Q}(A)\triangleq ...
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Differentials of Measures in the context of Radon Nikodym Derivatives.

I first note that a similar question was asked here: Calculating Radon Nikodym derivative, though the explicit steps used to calculate the derivative were not made clear. Over measurable space $([0,1]...
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$d\nu =f\,d\mu$ and $d\mu=g\,d\lambda$ implies $d\nu=fg\,d\lambda$

The full question is Let $\nu$ be a $\sigma$-finite signed measure, and let $\mu, \lambda$ be $\sigma$-finite positive measures on $(X, \Sigma)$. Show that $\nu \ll \mu$ and $\mu \ll \lambda \...
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What are sufficient conditions for the boundedness of a Radon-Nikodym derivative of a pull back measure?

Framework. Let $(X,\mu)$, and $(Y,\nu)$ be probability spaces on compact Hausdorff sets. Let $T:X\longrightarrow Y$ be a measurable function. Assume $\fbox{$T^{\ast}\mu\ll\nu$}$, where $T^{\ast}\mu$ ...
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What does the symbol ∟ mean?

I encountered this symbol $\lefthalfcup$ ∟ in the statement of Besicovitch derivation theorem. It says that the Radon-Nykodym decomposition of the given measure $\nu$ is $\nu=f\mu+\nu^s$, $\nu^s$ is ...
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$\|h\mu\|= \int |h|d|\mu|$

I was reading Conway and just popped in. I am trying to prove the note that part, but I cannot. Any help will be appreciated.
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Basic probability, Mass function, Density, CDF, Distribution, Random Variables.

Hi I'm hoping this will clear up any confusion on basic probability. Take a probability space $(\mathbb{P},\mathcal{F},\Omega)$ : $\Omega$ is some set of elements $\omega \in \Omega$ $\mathcal{F}$ ...
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Exact definition of Radon-Nikodym Theorem

A part of the RND Theorem states for $\sigma-$ finite measures $\nu, \mu$ on $(X,\mathcal{A})$ $\nu << \mu \iff \exists f$ non-negative and measurable: $\nu(A)=\int_{A}fd\mu$ for all $A \in \...
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Counterexample in Radon-Nikodym Theorem. Problem 38 Royden 2ed.

Use the following example to show that the hypothesis in the Radon-Nikodym Theorem that $\mu$ is $\sigma$-finite cannot be omitted. Let $X=[0,1],\ \mathcal{B}$ the class of Lebesgue measurable subsets ...
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Radon-Nikodym derivatives. Royden, Problem 33.

Radon-Nikodym derivatives. Show that that Radon-Nikodym derivative $[\frac{d\nu}{d\mu}]$ has the following properties: a. If $\nu\ll \mu$ and $f$ is a nonnegative measurable function, then $\int fd\...
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Confused on $ \in \mathcal{L}(X,\mu)$ and $\in \mathcal{L}(X,\nu)$

I have come across an uncertainty in a proof I have just read: Let $(X,\mathcal{A})$ be measure space and $\mu, \nu, \eta$ be $\sigma-$finite measures such that $\eta \ll \nu$ and $\nu \ll \mu$ Show ...
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Sigma Algebra Generated by Subset and Radon-Nikodym derivative

So this is a question to study for qualifying exams, not a homework question! Let (X,M,μ) be a measure space with μ(X) < ∞. Suppose A ∈ M and 0 < μ(A) < μ(X). Let N be the σ-algebra ...
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Radon-Nikodym derivative of a limit of measures with bounded R.-N. derivative

Let $(\Omega,\mathcal F)$ be a measurable space. Let $\lambda$ be a probability measure, and let $(\mu_k)_{k\geq 0}$ be a sequence of probability measures. Suppose that $\mu_k\ll\lambda$ for any $k$, ...
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Lebesgue-Radon-Nikodym decomposition

Please how to find the Lebesgue decomposition of $\nu$ with respect to the Lebesgue measure $m$ where $\nu$ is the Lebesgue-Stieltjes measure associated to the following function: $$ F(x) = \begin{...
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1answer
141 views

Radon-Nikodym Derivative of a Limit of a Sequence of Measures

Let $\nu$ be a sigma-finite measure, let $\mu$ be a measure absolutely continuous with respect to $\nu$, and let $(\mu_n)$ be a sequence of measures such that $\mu_n(E)\uparrow\mu(E)$ for all ...
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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 $\nu$...
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A counterexample to the epsilon-delta criterion for Absolute Continuity of Measures

Let $p>0$, and let $\mu$ be a Borel measure on $[0,\infty)$ defined by $\mu(E)=\int_Ex^pd\lambda$ where $\lambda$ denotes Lebesgue measure. Show that $\mu$ is absolutely continuous with respect to ...
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Understanding Proposition 3.9 Folland Part 2 (Radon-Nikodym Derivative)

Suppose that $\nu$ is a $\sigma$-finite signed measure and $\mu$, $\lambda$ are $\sigma$-finite measures on $(X,M)$ such that $\nu<<\mu$ and $\mu << \lambda$. a. If g $\in L_1(\mu)$ then ...
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Geometry of Banach Space (Radon Nikodym)

I read in David Williams, Probability with martingale : "The right context for appreciating the close inter-relations between martingale convergence, conditional expectation, the Radon-Nikodym theorem,...
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Radon-Nikodym derivative and its dynamic [closed]

Let the process $S_{t}$ under the historical measure P be as following $$dS_{t}=\mu S_{t}dt+V_{t}S_{t}dW_{t}^{1}$$ $$dV_{t}=(a+d-bV_{t})dt+\sigma(\rho dW_{t}^{1}+\sqrt{1-\rho^2}dW_{t}^{2})$$ Let Q be ...
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Finding a density for a LS-measure

I have a puzzling problem and hope some of you can give me a hint how to solve it. $F \geq 0$ is a distribution function of a LS-measure $\nu_F$. The following things are to do: (i) $G:=F^2$ is a ...
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Proofing that lower bound of two measures exists

I have some troubles with a problem about measures. This is the last missing subproblem of a bigger one, so I would be glad, if some of you can help me to finish this task. Consider two measures $\mu,...
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How to prove that Radón-nikodým theorem fails in this case?

In this exercise 8.M I have already shown that the collection of sets given forms an $\sigma$-algebra and that $ \lambda, \mu $ are measures and that $\lambda \ll \mu$. I could not show that the Radón-...
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Reweighting Radon Nikodym Derivative

Let $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space. I have a random variable on this space X such that $ess \ inf \ X < r < ess \ sup \ X$ where $r \in \mathbb{R}$. Now ...
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Assuming finite Radon-Nikodym to prove sigma-finite

Suppose that we assume the version of Radon-Nikodym for finite measure that is if $X$ is some set, $\mathcal{A}$ is a $\sigma$-algebra on $X$ and $\mu$ and $\nu$ are finite measures on $X$ where $\nu$ ...
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Radon-Nikodym derivative of mutually absolute continuous measures

Here is the problem I have. $\nu$ and $\mu$ are two positive $\sigma$-finite measures, such that $\nu \ll \mu$ and $\mu \ll \nu$. Also, the function $h$ is the Radon Nikodym derivative such that $h =...
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Radon Nikodym derivative

Given any two $\sigma-$finite measures $\mu$ and $\nu$ on some measurable space $(\Omega, \mathcal F)$ how do I compute $\dfrac{d\mu}{d(\mu+\nu)}$? If $\nu<<\mu$ then I could have said that $\...
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Question about Radon Nikodym

If $\mu$ is absolutely continuous with respect to $\upsilon$, then a theorem is $\frac{d|\mu|}{d\upsilon}=\left|\frac{d\mu}{d\upsilon}\right|$. Let's say that $\lambda$ is the Lebesgue measure and $\...
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Stochastic version of the Radon-Nikodým theorem

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a filtration on $(\Omega,\mathcal A)$ $\mathcal P$ denote the predictable $\...
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Does for two equivalent probability measures $Q \approx P$ boundedness in $L^1(Q)$ imply boundedness in $L^0(P)$?

Assume we have a measurable space $(\Omega,\mathcal{F})$ with two probability measures $Q$ and $P$, which are equivalent, i.e. for all $A \in \mathcal{F}$ we have $$Q(A)=0 \iff P(A)=0.$$ I will ...
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Expression for the density function of a smooth function

I am working on tomographic methods in which the data is the "distribution" of values along a line rather than an integral. Given a measurable function $f:[0,1] \rightarrow \mathbb{R}$ one can define ...
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Calculating Explicit Radon-Nikodym Derivatives

I am stuck on the following problem: Let $f(x,y) = \max \{ x^2 + y^2 , 1 \}$ and define a Borel measure $\mu$ on $\mathbb{R}$ by $\mu(E) : = (m \times m)(f^{-1}(E))$, where $m$ is Lebesgue measure. ...
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1answer
51 views

Radon-Nikodym property

I am currently struggling with the following: Let $\nu,\mu$ be finite measures on a measurable space $(\Omega,\mathcal{A}) ~$with$~ \nu \ll \mu$ and $f:=\frac{d\nu}{d(\mu+\nu)}$. I would like to show ...
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Is there an abstract/generalized way to consider and calculate mappings of random variables that is continuous and 'monotonic' but non-linear?

I am doing a lot of work with lognormal RVs. I am trying to get my head around the formal mathematics of the non-linear transform of a random variable, particularly where there isn't any '...
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Importance of $\sigma$-finiteness for Uniqueness of Measure $dm_f = f dm$

Background Given a measure $m : X \rightarrow [0,+\infty]$ and a measurable function $f : X \rightarrow [0,+\infty]$, we can define a new measure by integrating $f$ as below: $$ m_f(E) = \int_E f \, ...
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1answer
102 views

Radon–Nikodym derivative with another variable is a martingale

Let $N > 0$ be an integer. Suppose $Y$ is a random variable on $\Omega$ and define, for $n = 0, 1, \ldots, N$, $$ Y_n = \tilde{E}_n(Y) $$ (i.e. the conditional expectation with respect to the risk-...
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Change of measure as functions of random variable

Let $X$ and $Y$ be indepdendent under the probability measure $P$. Define two equivalent probability measures $Q$ and $R$ by $$\frac{dQ}{dP}=g(X), \frac{dR}{dP}=g(X)h(Y)$$ where $g, h : \mathbb{R} \...
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58 views

Can divergence be thought of as a Radon-Nikodym derivative?

The divergence theorem states roughly $$\int_\Omega \operatorname{div}U\ dV=\int_{\partial \Omega}U\cdot n\ dS$$ where $U$ is a vector field, $\Omega$ is a region of space with a smooth boundary, $n$...