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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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1answer
162 views

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 ...
2
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1answer
21 views

$\|h\mu\|= \int |h|d|\mu|$

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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18 views

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 $\omega \in \Omega$ $\mathcal{F}$ ...
0
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0answers
32 views

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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2answers
32 views

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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0answers
37 views

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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0answers
35 views

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 ...
1
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1answer
23 views

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 ...
2
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2answers
39 views

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 $k$, ...
0
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2answers
29 views

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
78 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 ...
0
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2answers
59 views

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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2answers
76 views

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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3answers
68 views

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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0answers
24 views

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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0answers
18 views

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 ...
0
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1answer
11 views

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 ...
1
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1answer
31 views

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,...
2
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0answers
24 views

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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0answers
37 views

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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0answers
49 views

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 $\nu$ ...
0
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1answer
38 views

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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0answers
67 views

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 $\...
1
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1answer
33 views

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 $\...
1
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1answer
46 views

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 $\...
3
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1answer
40 views

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 ...
0
votes
0answers
48 views

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 ...
1
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1answer
83 views

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. ...
0
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1answer
45 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 ...
0
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0answers
8 views

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 '...
2
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0answers
34 views

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 \, ...
2
votes
1answer
76 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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0answers
39 views

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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0answers
48 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, $n$...
4
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0answers
37 views

Deriving the posterior distribution given arbitrary measures

I am given the following facts: $$ \forall A \in \mathcal{B}: \; \mu(A) = \int_{A} f(x) \alpha(\text{d} x), \quad \text{ and } \quad \nu_x(A) = \int_{A} g_x(y) \beta(\text{d} y), \; \forall x \in \...
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2answers
54 views

Lebesgue-Radon-Nikodym representation

In Folland, $\textbf{3.22 Theorem.}$ Let $\nu$ be a regular signed or complex Borel measure on $\mathbb{R}^n$, and let $d\nu = d \lambda + f dm$ be its Lebesgue-Radon-Nikodym representation. Then ...
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1answer
57 views

The Dual of $L^{\infty}$ and the Radon-Nikodym Theorem

I've run into an issue with an exercise and I think that more assumptions are needed in order for the result to be true. Here is the exercise: Let $\nu:\mathcal{M}\to\mathbb{R}$ be a signed measure ...
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1answer
62 views

Finding the Radon-Nikodym Derivative

How does one go about finding the Radon-Nikodym derivative? I've seen the following exercise. Let $(\mathbb{N}, \mathcal{P}(\mathbb{N}), \mu)$ be a measure space, where $\mu$ is the counting measure....
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1answer
58 views

Radon-Nykodym Derivative process-Property of Conditional Expectation

I'm struggling with the understanding of the properties of a Radon-Nykodym Derivative process. In my example we defined the probability measure $\mathbb{Q}[A]=\int_A Z(\omega) d\mathbb{P}(\omega)$. ...
4
votes
1answer
133 views

Radon Nikodym derivative $\frac{\mathrm d(fλ)}{\mathrm d(gλ)}$

Here is the question: Consider the space $X = [0,1]$ with Lebesgue measure $\lambda$. Let $\mu = f\lambda$ and $\nu = g\lambda$ with functions $f$ and $g$ nonnegative, be finite measures. Find a ...
1
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1answer
37 views

Application of chain rule, is this correct?

I have $\sigma$ finite measures $\lambda$ and $\mu$ on the same space $(X, \Sigma)$, and $\nu = \lambda + \mu$. An exercise I have is to show that $\lambda \ll \mu$ if and only if $$\nu(\{d\mu/d\nu(x)...
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0answers
105 views

What is the Radon-Nikodym density $dP^∗ /d P$ of the unique $P^ ∗ ∼ P$ such that the discounted price $S^*_t := S_t /B_t$ is a $P^∗$-martingale

Consider the Black-Scholes Model where we have the following risky asset $dS_t = \mu S_t dt + \sigma S_t dW_t, t\in[0,T] t≥0 ,S_0 = s >0 $ where $\mu,\sigma$ are positive constants and a ...
1
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1answer
83 views

ergodic measure and absolutely continuous measure

i've come across this problem in Petersen's "Ergodic Theory": Let $(X,\mathcal{B},T,\mu)$ be an ergodic dynamical system. Let $\nu\ll\mu$ be a measure un $(X,\mathcal{B})$ such that $\nu T^{-1}\ll\nu$...
2
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1answer
60 views

Show that the Radon-Nikodym density is $\mathcal{G}$ measurable iff $E_P[X|G] = E_Q[X|G]$

Let $\alpha$ and $\beta$ be equivalent probability measures on $(\Omega, \mathcal{F})$, with Radon-Nikodym density of $\alpha$ wrt $\beta$ is $\eta$, i.e., for all $A \in \mathcal{F}, \beta(A) = \...
2
votes
2answers
62 views

Computing expectation under a change of measure [duplicate]

Let $X$ be a random variable on a probability space $(\Omega,\mathscr F, P)$. Define a new probability measure $$\tilde P(A) = E[1_A X]$$ for all $A\in\mathscr F$. Let $\tilde E$ be expectation taken ...
0
votes
2answers
55 views

Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $ [closed]

How do I prove the following? I don't know where to start. If $X$ is a random variable with $E^{\mathbb P}[X] = \mathbb P(X>0)=1$ and $ \mathbb Q$ is the probability measure defined by $ \...
3
votes
1answer
164 views

Application Radon - Nikodym

I just learned about Radon-Nikodym theorem. However, I do not seem to have any intuition on how to apply it... For example : Let $(X,\mathscr{M},\lambda)$ be a $\sigma-$finite measure. Let $f$ be $\...
4
votes
0answers
115 views

Girsanov theorem and filtrations

Let $\{W_t\}$ be a standard Wiener process on a probability space $(\Omega, \mathcal{F},P)$. Let $\mathcal{F}^W$ be the natural filtration generated by $\{W_t\}$. Let $\{\theta_t\}$ be an $\mathcal{...
3
votes
1answer
764 views

Finding Radon-Nikodym derivative

Let $m$ be Lebesgue measure on $\mathbb R_+=(0,\infty)$ and $\mathcal A = \sigma\left(( \frac 1{n+1} , \frac 1n ]:n=1,2,...\right)$. Define a new measure $\lambda$ on $\mathcal A$, for each $E \in \...
3
votes
0answers
203 views

Specific Radon-Nikodym Derivative Interpretation

Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says that if $P$ is absolutely continuous with respect to $Q$, then there exists ...