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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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Radon-Nikodym theorem, integrability

$\textbf{Theorem 4.3}$ Suppose $\mu$ is a $\sigma$-finite positive measure on the measure space $(X, \mathcal{M})$ and $\nu$ a $\sigma$-finite signed measure on $\mathcal{M}$. Then there exist unique ...
the topological beast's user avatar
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Radon-Nikodym derivatives with restricted support

Let $\lambda$ be the Lebesque measure on $\mathbb{R}$ and $f$ and $g$ be two PDFs of two random variables. Then we can write: $$f(x) = \frac{d \mu}{d \lambda}(x) \qquad g(x) = \frac{d \nu}{d \lambda}(...
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Show that the $\sigma$-finiteness assumption of $\mu$ cannot be omitted in Radon-Nikodym Theorem

The Radon-Nikodym Theorem says the following: Theorem$\quad$ Let $(X,\mathscr{A})$ be a measurable space, and let $\mu$ and $\nu$ be $\sigma$-finite positive measures on $(X,\mathscr{A})$. If $\nu$ ...
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Invertibility of a measurable mapping from lower and upperbounds on the induced pushforward measure

Let $\Omega \subseteq \mathbb{R}$ be open and consider the standard Borel space $(\Omega, \mathcal{B}(\Omega), \mu)$, where $\mu$ denotes the Lebeasgue measure. Let $f: \Omega \to \Omega$ be a ...
Saleh's user avatar
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Confused about the Radon-Nikodym theorem applied to the counting measure

Sorry, I just need someone to clear up a confusion I have. Let $(X,\mathcal{F},\mu)$ be any measure space, and let $\#$ be the counting measure on $X$. We have that $$\#(A) = 0 \implies |A| = 0 \...
Sam's user avatar
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Proving that "a distribution has a density function if and only if its cumulative distribution function $F(x)$ is absolutely continuous".

Wikipedia states that A distribution has a density function if and only if its cumulative distribution function $F(x)$ is absolutely continuous. What is the exact statement of the result above? It ...
Sam's user avatar
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3 votes
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Radon Nikodym derivative and distribution function

Let $\mathfrak{B}$ be the Borel $\sigma$-algebra over $\mathbb{R}$ and $\beta$ the Borel-Lebesgue measure over $\mathfrak{B}$. Let $\mu$ be a Borel measure over $\mathbb{R}$ s.t. the distribution ...
Kham Bodrogi's user avatar
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10 views

Why $d\mu (q)\delta (k,q)$ is $G$-invariant?

Let $G$ be a Lie group acting transitively on a smooth manifold $M$ endowed with a quasi-invariant measure $\mu$ (then there exists Radon-Nikodym derivative $\rho_f$ for every $f\in G$). For $k\in M$,...
Mahtab's user avatar
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Which Probability Distributions Dominate the Lebesgue Measure?

Recall In probability theory, the distribution $\mu_X$ of a random variable $X$ (on some unspoken probability space) refers to the measure $\mu_X(A) := \mathbb{P}(X \in A)$ that is defined on the ...
Thomas Winckelman's user avatar
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Radon Nikodym derivative for the sum of two measures

Suppose that $\mu$ and $\nu$ are two $\sigma$-finite measures and let $\rho=\mu+\nu$. Obviously $\mu$ and $\nu$ are both absolutely continuous w.r.t. $\rho$. I am trying to find a closed form ...
Florian Brück's user avatar
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Integration over measure which counts jump discontinuities of specific length.

$f:[0,1] \to [0,1]$ is a monotonically increasing function with $f(0)=0$ and $f(1)=1$. Let $p>1$. Define, for $0\leq a < b \leq 1$, $\mu((a,b])$ as the number of points $x \in [0,1]$ such that $\...
Another_Ramanujan_Fan's user avatar
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1 answer
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Convergent sequences under different probability measures

What is the „intuitive” reason behind the following statements... Let $\left(X_{n}\right)_{n}$ be a sequence of random variables. Let us assume $\mathbf{Q}\ll\mathbf{P}$, i.e. the $\mathbf{Q}$ ...
Kapes Mate's user avatar
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Unique decomposition of complex Borel measure

I need to prove that the following decomposition of complex Borel measure is unique. $$\mu=\mu_d+\mu_{a c}+\mu_{sc}$$ Here $\mu_d$=discerete measure, $\mu_{ac}$=absolutely continuous measure, $\mu_{sc}...
Siddharth Prakash's user avatar
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Equality of the $L^1$ norm and measure norm of an corresponding absolutely continuous measure for BV functions

Let $u \in BV(I;\mathbb{R}^d)$ be a vector valued function of bounded variation, (in particular Bochner integrable) on an open interval $I \subset \mathbb{R}$. We can then define the Radon measure $u \...
ThommyAC's user avatar
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How do we deduce that a measure is absolutely continuous with respect to Lebesgue?

This question stems from [The kinetic limit of a system of coagulating Brownian particles] (https://arxiv.org/abs/math/0408395), specifically the last step in the proof of Lemma 4.2. The setting is as ...
Florian Ente's user avatar
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show a measure is absolutely continuous wrt lebesgue

Take a measure $\mu$ on $[0,T]\times \mathbb R^d$ and a positive mollifier $\eta^\delta$ on $\mathbb R^d$. Assume for all $\gamma :[0,T]\to \mathbb R_+$ with $\int_0^T \gamma(t)dt=1$ we have \begin{...
Florian Ente's user avatar
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Doubts about Radon-Nikodym Theorem

I'm learning Radon-Nikodym theorem and I have some doubts now. Let $\mu(B(x,r))=r^2$ for each $x\in\mathbb{R}$ and $r>0$, then $\mu$ is a radon measure on $\mathbb{R}$ and $\mu<<\mathscr{H}^1$...
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Distribution induced simultaneously by a local integrale function and a radon measure

Let $\Omega$ be an open subset of $\mathbb{R}^n$. If $\mu$ is a Radon measure on $\mathbb{R}^n$, then we denote with $T_\mu$ the distribution defined as follow: $$<T_\mu, \phi>=\int_\Omega \phi ...
Shiva's user avatar
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Reference Request: proof that R-N Derivative is a ratio of PDFs for any two RVs on $\mathbb{R}$

Consider any two continuous random variables defined on the Borel-measurable sets on $\mathbb{R}$. I would like to reference a proof of the following statement: "The distributions of the two ...
Jan Stuller's user avatar
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Radon-Nikodym Derivative for paths of stochastic process

I am currently trying to understand this paper https://arxiv.org/pdf/1505.07612.pdf and have problems understanding lemma 3.1. We have a stochastic Process given by \begin{equation} ...
kays44's user avatar
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If $\nu_n \ll \mu$ and $|\nu_n - \nu|(X) \rightarrow 0$ for finite measures $\mu, \nu_n, \nu$ on space $X$, then does $\nu \ll \mu$?

This is a question from here: Let $\mu, \nu, \nu_n$ be finite measures on $(X, \mathcal{F})$, and suppose that $\nu_n \ll \mu$ for all $n \in \mathbb{N}$. Is the following true or false? If $|\nu_n - ...
HorribleATMath's user avatar
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definition of singular measure

Lebesgues Decomposition Theorem says that if $\mu$ and $\nu$ are two $\sigma$-finite measures, $\mu$ can be decomposed into $\mu=\mu_{ac}+\mu_{sing}$ where $\mu_{ac}$ is absolutely continuous with ...
Kiko's user avatar
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Let $\mu$ and $\nu$ be two measures on $(X,\mathcal{M})$. Show that there is measurable $f:X\rightarrow[0,1]$ such that $\int_A(1-f)d\mu=\int_Afd\nu$

We also know these two measures are finite. The hint to this question says to use the Radon-Nikodym theorem, but it seems that either they are referencing an alternate phrasing that I cannot find, or ...
cable's user avatar
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1 answer
132 views

Show that a measure is absolutely continuous w.r.t Lebesgue and an inequality holds

I have been struggling with this question for almost a week, which was one of the questions in a past exam. I would really appreciate it if anyone could help me with this problem. Let $\mathcal{M}$ be ...
Harry's user avatar
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Radon-Nikodym property of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space and $B(\mathcal{H})$ the space of bounded linear operators mapping $\mathcal{H}$ to $\mathcal{H}$. Does $B(\mathcal{H})$ have the Radon-Nikodym property? ...
crimsonmist's user avatar
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1 answer
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Unicity in Radon-Nikodym Theorem

Folland states that if $\nu$ is a $\sigma-$finite signed measures and $\mu$ is a $\sigma-$finite measure we have: $$\nu=\nu_c+\nu_s \quad \text{such that $\nu_c<<\mu$ and $\nu_s \perp \mu$.}$$ I ...
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1 answer
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Change of measure for measures on different spaces

For two measures $\nu, \mu$ that are defined on the same measure space with $\nu = \int f d\mu$, it is a well known result, mostly used in the context of the radon nikodym theorem, that for an ...
guest1's user avatar
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Transforming Radon Nikodym derivatives

I am currently confused by the (abuse of?) notation regarding the Radon Nikodym derivative in many proofs. What I am currently struggling with in particular is a proof of the classical information ...
guest1's user avatar
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Characterization of measurable G such that $G \circ Z$ has a density with respect to Lebesgue Measure

Assume Z is a real valued random variable such that it's distribution has a density with respect to Lebesgue Measure (i.e. its Radon-Nikodym derivative w.r.t Lebesgue measure exists.) Then what is the ...
JPomegranate's user avatar
3 votes
1 answer
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Radon-Nikodym for thress measures.

Let $\mu_1$ and $\mu_2$ and $\nu$ be two finite measures on the measurable space $(\Omega, \mathcal{S})$ such that $\nu \ll \mu_i$, $i = 1, 2$, and $f_i = \frac{d\nu}{d\mu_i}$. Show that then $\nu \ll ...
vwhg1050's user avatar
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Radon-Nikodym chain rule computation trouble

I'm having trouble figuring out where my error is for the following computation. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$, and define the measure $\mu$ via $\mu(A) = \lambda(A^3)$. ...
AMACB's user avatar
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Understanding the consequence of Radon-Nikodym Theorem

In Rudin's RCA it says the following: Suppose $\mu$ is a complex measure on a $\sigma$-algebra $\mathfrak{M}$ in $X$. Then there is a measurable function $h$ such that $|h(x)|=1$ for all $x\in X$, ...
MathLearner's user avatar
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1 answer
111 views

Can the linear space of all finite signed measures be put an inner product so that $\mu\bot\nu$ exactly when $\mu$ and $\nu$ are mutually singular?

In Folland's "Real Analysis", two signed measures $\mu,\nu$ on a measurable space $(X,\mathcal{M})$ are said to be mutually singular if there is a decomposition of $X=E\cup F$ so that $E$ is ...
Tiffany's user avatar
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Construction of a closed subset of irrationals

Let $S=[0,1]\setminus \mathbb{Q}$ and let $A\subset S$ be a closed subset with a positive Lebesgue measure ($=\mathcal{L}(A)>0$). I want to show the following. Let $b\in[0, \mathcal{L}(A))$ then ...
math_as_a_lifestyle's user avatar
3 votes
0 answers
160 views

Proof about Markov kernels and absolute continuity

Assumptions: $(\mathsf{X}, \mathcal{X})$ is a measurable space. $M_n$ and $L_{n-1}$ are Markov probability kernels for $n=2, \ldots, P$. $\mu_n$ be probability measures on $(\mathsf{X}, \mathcal{X})$ ...
Physics_Student's user avatar
2 votes
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Behavior of Radon-Nikodym derivative sequences dependent on f and $\chi^2$ Divergence

First, let's consider the distribution series: $\left\{ \mu _m \right\} _{m\geqslant 1}$ and $\left\{ \nu _m \right\} _{m\geqslant 1}$. They are all defined in the bounded subset $\Omega$ of $\mathbb{...
Sizhe Ding's user avatar
3 votes
1 answer
363 views

Is the Converse of the Radon-Nikodym Theorem true?

I'm curious as to whether the converse of the Radon-Nikodym theorem holds: Converse Theorem: let $\mu.\nu$ be measures on $(\Omega,\Sigma)$ such that $$\nu : A\mapsto \int_A f \ d\mu$$ for some ...
Sam's user avatar
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1 answer
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How to recover classical derivative from Radon-Nikodym derivative

Let $\mu$ be an unsigned measure which is differentiable with respect to the Lebesgue measaure $m$. Let $f$ be its Radon-Nikodym derivative w.r.t. $m$. i.e. $f=\frac{d\mu}{dm}$, or alternatively $\mu(...
quanticbolt's user avatar
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2 votes
1 answer
117 views

Radon-Nikodym derivative of pushforwards: $\frac{d f_\# \mu}{d g_{\#} \mu}$

Let $f, g \colon (0, 1) \to \mathbb R$ be two functions (both spaces are equipped with their respective Borel $\sigma$ algebras). What is the Radon-Nikodym derivative of $f_{\#} \lambda$ with respect ...
ViktorStein's user avatar
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1 vote
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Two attempts to define a conjugate of a complex measure - how are they related?

I am slightly confused about two different approaches to arrive at a conjugate of a complex measure. Any hints and/or clarifications on how these notions are related would be great. (It is also ...
P.Jo's user avatar
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2 answers
164 views

Help Understanding Measure Theory definitions of expectations and densities

This is a bit of a continuation of my last post - I'm studying measure theory and probability and am trying to relate the definitions of expectations, pdf's and cdf's from a typical first year ...
Jamal's user avatar
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2 votes
1 answer
260 views

Folland's real analysis, Dirac $\delta$-function as the Radon-Nikodym derivative

In Folland's real analysis, it writes Nonexample: Let $\mu$ be Lebesgue measure and $\upsilon$ the point mass at 0 on $(\mathbb{R}, \mathcal{B}_{\mathbb{R}})$. Clearly $\upsilon \perp \mu$. The ...
sam2018's user avatar
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Help Understanding Expectations, density and Radon Nikodym Theorem

I am trying to study probability and measure theory on my own, and I'm struggling to make the link between the Law of a Probability (distribution), the Radon Nikodym Theorem and the change of ...
Jamal's user avatar
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0 answers
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Finding the Radon-Nikodym derivative (a particular instance)

Consider a measurable space $\left( E, \mathcal{E}, \mu \right)$, where $\mu$ is $\Sigma$-finite, i.e., there is a sequence $(\mu_n)_{n=1}^{\infty}$ of finite measures on $\left( E, \mathcal{E} \right)...
Fran712's user avatar
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15 votes
1 answer
269 views

What is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure?

The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $...
fromscratch's user avatar
2 votes
1 answer
183 views

Radon–Nikodym derivative Relation to derivative in calculus?

I am reading chapter 3 of Folland's Real Analysis Book, where it introduces the Radon-Nikodym Derivative of a sigma finite signed measure with respect to another sigma finite positive measure. Let $\...
Remu X's user avatar
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Calculating a concrete Radon-Nikodym derivative (density)

I am learning measure theory through self-studying and am trying to solve an exercise related to the Radon-Nikodym derivative. Specifically, Exercise 1.4.3 on page 57 in The Theory of Statistics and ...
saper0's user avatar
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2 votes
2 answers
371 views

Change of variables from Radon–Nikodym theorem

Assuming the usual required conditions (https://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem#Radon%E2%80%93Nikodym_theorem), the Radon-Nikodym theorem states: There exists a measurable function ...
Scott Hahn's user avatar
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323 views

Decomposition of probability measures with a bounded total variation distance

Fix a probability space $(\Omega, \mathcal{F})$. Let $P$ and $Q$ be two probability measures on $(\Omega, \mathcal{F})$ such that there exist $\varepsilon$ and $\delta$ that for every $A \in \mathcal{...
MMH's user avatar
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1 answer
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property of expectation when $\Bbb{Q} \ll \Bbb{P}$

If $\Bbb{P}$ and $\Bbb{Q}$ are two probability measures such that $\Bbb{Q} \ll \Bbb{P}$ (i.e $\Bbb{Q}$ is absolutely continuous with respect to $\Bbb{P}$), can something be said about the expectation ...
yrual's user avatar
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