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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Converse to Lebesgue Differentiation Theorem

Suppose we are given two finite positive Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$ such that the function $$ f(x) := \limsup_{r\to 0} \frac{\mu(B(x,r))}{\nu(B(x,r))} $$ is in $L^1(\nu)$. Is it ...
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1 vote
1 answer
27 views

Conditional density: exact definition

I have trouble understanding the connection of conditional densities for two (continuous) random variables and conditional distributions for measurable sets. Let $\nu$ be a probability measure on $\...
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1 vote
1 answer
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Is it a problem that we have to *choose* reference measure in Likelihood function?

Suppose we have some parametric model $\{P_\theta \ \colon \theta \in \Theta\}$ and a sample $X$. If we suppose as usual in classical statistics that $P_\theta << \lambda$ for all $\theta \in \...
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  • 705
2 votes
1 answer
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Total variation of a complex measure and Radon-Nikodym derivative

In Folland's text he states, The total variation of a complex measure $\nu$ is the positive measure $|\nu|$ determined by the property that if $d\nu = f d\mu$ where $\mu$ is a positive measure, then $...
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Radon–Nikodym derivative of two multivariate Gaussians

Let two Gaussian distributions $P_1$, $P_2$ with mean $0$ and covariance matrix $\boldsymbol{\Sigma}_1,\boldsymbol{\Sigma}_2\in\mathbb{R}^d$ be given. I want to calculate the Radon–Nikodym derivative ...
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2 answers
22 views

Relationship between radon-nikodym derivatives and total variation distance

Let $(\mathcal{X},\mathcal{A})$ be a measure space on which we have defined two probability measures $P$ and $Q$. Let $f$ and $g$ denote Radon-Nikodym derivatives of $P$ and $Q$ with respect to a $\...
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1 vote
1 answer
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Question on relation between radon-nikodym derivatives and the total variation distance

Let $(\mathcal{X},\mathcal{A})$ be a measure space on which we have defined two probability measures $P$ and $Q$. I am reading some notes online which makes the following jump without explanation in ...
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4 votes
0 answers
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Radon-Nikodym derivative of pushforward measures and Girsanov theorem

Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
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Is the Radon-Nikodyn derivative unique (under a few assumptions)

For two measures $\mu$ and $\nu$ (with $\mu \ll \nu$) over a topological space $X$, $\mu$ can be expressed as $$\mu(A)= \int_A \frac{d\mu}{d\nu}d\nu$$ and the function $\frac{d\mu}{d\nu}$ is the Radon-...
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  • 352
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1 answer
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Change of probability measure (something relative to Brownian motion)

I'm working on a question, it says assume that $X_1, ..., X_n$ are i.i.d. random variables with the finite moment generating function $E[e^{\theta X_1}]$ (under the measure P). Define $$\kappa(\theta)=...
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1 answer
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Orthogonality of joint probability and conditional probability measures

Suppose that $(X,Y)$ are real valued random variables on some space $(\Omega, \mathcal{F})$. Let $P,Q$ be two possible joint probability measures for $(X,Y)$. Let $P_{Y|X}$ and $Q_{Y|X}$ be two ...
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1 vote
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Martingale derived from Radon-Nikodym derivative - why is the supremum of the martingale finite?

I have come across a statement I don't understand while reading the following post :https://almostsuremath.com/2010/05/03/girsanov-transformations/ Given a probability measure $\mathbb{P}$, we have an ...
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1 vote
1 answer
81 views

Products of Extended-Integrable Functions are Extended-Integrable

Here I state the proposition: Let $(X, \mathcal{F}, \mu)$ be a measure space and $f: X \to \mathbf{R}$ be a $\mu$-semi-integrable function. Then there exists a signed measure $\lambda: \mathcal{F} \...
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6 votes
0 answers
146 views

$G$ acts transitively on a space $X$. If a function on $X$ is $G$-invariant up to measure zero, is it necessarily a constant (up to measure zero)?

Consider a locally compact Hausdorff $σ$-compact topological space $X$ and a locally compact Hausdorff $σ$-compact topological group $G$ acting continuously and transitively on $X$ such that there ...
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2 votes
1 answer
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Additivity of Radon-Nikodym derivatives of signed measures

In the section on the Lebesgue-Radon-Nikodym theorem, Folland's Real Analysis: Modern Techniques and Their Applications (Second Edition) pp. 91 says that "it is obvious that $d(\nu_1 + \nu_2)/d\...
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The (upper/lower) derivative of a locally finite Borel measure w.r.t. another such measure is a Borel-function.

Let $\mu$ and $\lambda$ be locally finite Borel measures on $\mathbb{R}^n$. We may define an upper resp. lower derivative of $\mu$ w.r.t $\lambda$ at $x\in\mathbb{R}^n$ by $$ \underline{D}(\mu,\lambda,...
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1 vote
2 answers
101 views

If $\int_A f d \mu = \int_A g d\mu$ for continuous positive $f,g$ and a Radon measure $\mu$, is $f=g$ a.e.?

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$. If $f,g: X \to (0, \infty)$ are continuous functions such that $$\int_A f d \mu = \int_A g d\mu$$ for all Borel subsets $...
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2 votes
1 answer
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Prove that $d\mu/d\nu$ is "measurable" in $\mu$

Question. Let $\lambda$ denote Lebesgue measure on $\mathbb R^d$. Rudin shows that for any $x\in \mathbb R^d$, any "regularly" decreasing sequence of Borel sets $E_0(x) \supset E_1(x) \...
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  • 587
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an equality of integrals on measure spaces with a measure and its image measure [duplicate]

Let f : X $\rightarrow$ $\mathbb{R}$ be a measurable function on a measure space (X, $\Sigma$, $\mu$). Then the formula $\nu$(B) = $\mu$(f-1[B]) defines a Borel measure on $\mathbb{R}$. Prove, that $\...
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7 votes
1 answer
183 views

Does Radon-Nikodym derivative affect the Variance of a Random Variable?

Edit 26 January 2022: The answer below elegantly shows that when the diffusion term of an Ito process is not constant in $\omega$, it is generally not true that Variance remains unaffected by the ...
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2 votes
1 answer
77 views

Limit of integral exists imply a sequence of function converges in $L^1$

Let $\{f_n\}$ be a non-negative sequence in $L^1(X,\mathcal{M},\mu)$ with $\mu(X) < \infty$ satisfying, for each $E\in \mathcal{M}$, $$ \lim_{n\to\infty}\int_E f_n d\mu $$ exists, and $\{f_n\}$ ...
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2 votes
0 answers
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The Radon-Nikodym derivative for a sequence of dependent variables

Suppose that a probability space $(\Omega, \Sigma, \mathbb{P})$ is given. Let $W=\{W_n\}_{n\in \mathbb{N}_0}$ be a sequence of $\mathbb{P}$-i.i.d real-valued random variables on $\Omega$. Furthermore, ...
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0 votes
1 answer
64 views

Kullback-Leibler divergence for push forward measures

I have some trouble understanding a step in the proof of Lemma 2.1 from [1]. I believe that it is not supposed to be very hard as it is not really justified in the paper, but I am not very familiar ...
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9 votes
2 answers
193 views

Rigorous definitions of probabilistic statements in Machine Learning

In a supervised machine learning setup, one usually considers an underlying measurable space $(\Omega, \mathcal{F}, \Bbb P)$ and random vectors/variables $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \...
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1 vote
0 answers
105 views

Folland Theorem 3.22

At the start of the proof for Theorem 3.22, Folland says that $dv=d\lambda + fdm$ implies $d|v|=d|\lambda| + |f|dm$. I get why this is the case for positive and signed measures, but I'm not sure how ...
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2 votes
1 answer
63 views

Simple question about Radon-Nikodym derivative/integrating wrt an integral

This relates to the top answer here: Question on integral, notation and Nikodym derivative Suppose that $\nu << \mu$. Then we can find a non-negative $f$ s.t. $$\nu(E) = \int_{E} d\nu = \int_{E} ...
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  • 305
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1 answer
59 views

Change of measure defined by Radon-Nikodym derivative

Suppose $\{X_t\}_{t\geq0}$ is a nonnegative discrete-time martingale with $X_0=1$. Then we know by martingale convergence theorem that $X_\infty=\lim_{n\to\infty}X_n$ exists. Let $\mathcal{F}_n$ be ...
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1 vote
0 answers
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Is Absolute Continuity can be inferred from an Atomless Probability Space?

I have not deep knowledge in measure theory, and I am wondering if one could help me with this question. Based on what is defined in Wikipedia, It turns out that non-atomic measures actually have a ...
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2 votes
0 answers
30 views

Finding the lebesgue decomposition of a piecewise continuous function $F(x)$

Define $F(x) = \begin{cases} 0 & x < -1 \\ x + 2 & -1\leq x < 0 \\ x^2 + 3 & 0\leq x < 1\\ 6 & x \ge 1 \end{cases}$ I want to ...
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  • 1,806
1 vote
0 answers
48 views

When will Radon Nikodym derivative equals zero

Let $\mathcal{P}$ and $\mathcal{Q}$ be two $\sigma$-finite measures on the measure space $(\Omega,\mathcal{F})$. Let $\mathcal{Q}$ be absolutely continuous with respect to $\mathcal{P}$, that is, for ...
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  • 63
0 votes
1 answer
37 views

Radon Nikodym derivative countably additive? Infinite sum of absolutely continuous measures

Let $\mu, v_n$ be measures on $(X, \mathcal{B})$ with $\mu$ being $\sigma$-finite and $sup\{v_n(X)\} < \infty$. Note that $v = \sum_{n \ge 1} 2^{-n} v_n << \mu$ if and only if $v_n << \...
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  • 1,806
2 votes
1 answer
99 views

Radon-Nikodym derivative and derivative of functions of real variable

I have read Radon–Nikodym derivative and "normal" derivative but I am still quite confused. Here are the two kinds of derivative and I am looking for the analog: The derivative of a ...
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  • 195
1 vote
0 answers
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Is the ball ratio theorem for Radon Nikodym derivative known for general metric spaces?

Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu<<\mu$, it is well known that $$\frac{d\nu}{d\mu}(x)= \lim_{\epsilon\to 0} \...
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4 votes
2 answers
73 views

Deduce that $Z_m$ is the Radon-Nikodym-density $Z_m=\frac{dQ_m}{dP}$ of the probability measure $Q_m$, which is equivalent to P

The information given: Consider an arbitrage-free one-period financial market model $(S^0;S)$. We have a risk free asset $S^0$ with $S_0^0=1,S_1^0=1$ so the risk free rate $r=1$. We have a risky asset ...
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  • 83
0 votes
1 answer
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Radon-Nikodym Equivalent Measures

Problem statement: For a measure space $ (\Omega, \mathcal{F}, \mathrm{μ})$ : ${μ} $ is positve and finite, show the following: The set function $\nu(A) = \int_Af d\mu$ , $A \in \mathcal{F}$ defines ...
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2 votes
0 answers
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How to calculate the Doléans-Dade exponential/Radon–Nikodym density given an SDE of a defaultable bond?

Roughly speaking, when we have the dynamics of a stock process be given by the SDE: $$ dS_t = \mu_tS_tdt + \sigma_tS_tdW_t,$$ and if we can choose a cash account $B_t = e^{\int_0^t r_s ds}$ as ...
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3 votes
1 answer
43 views

Show that $\frac{dP}{d\nu} = \sum_{n=1}^{\infty} a_n \frac{d P_n}{d \nu}$?

Let $P=\sum_{n=1}^{\infty} a_n P_n$ be a probability measure where $a_n >0$, $\{P_n\}$ be a sequence of probability measures and $\sum_{n=1}^{\infty} a_n=1$. If $P \ll \nu$ where $\nu$ is a sigma-...
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0 votes
0 answers
42 views

Density w.r.t. measure on the unit simplex

Let $M$ be a measure on the unit simplex $S_d=\{ \mathbf{x} \in \mathbb{R}_+^{d}: ||\mathbf{x}||_1=1 \}$ where $||.||_1$ is the $\mathcal{L}_1$ norm and $\mathbb{R}^d_+=[0, \infty)^{d}$. A text I am ...
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  • 397
7 votes
2 answers
337 views

Conceptual Issues in the Measure Theoretic Proof of Conditional Expectations (via Radon-Nikodym)

I have been looking into measure theory (from a probabilist's perspective), and I have found the proof of the existence of the conditional expectation to feel a little "glossed over" in ...
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0 votes
0 answers
54 views

delta function as density w.r.t. counting measure

As we know, the delta function is not a Radon-Nikodym density with respect to the Lebesgue measure. If we choose the counting measure $\mu$, which assigns to every set the number of its elements, then ...
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1 vote
1 answer
28 views

Change in expectation due to Bayesian update

Consider a prior $P$ on some finite set $X$, and a posterior $Q$ on $X$ formed via a Bayesian update of $P$ given some data $y$. I am interested in finding alternative ways to write the following ...
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0 votes
1 answer
67 views

To show absolute continuity of measures.

Let $\delta_x$ denote the measure defined by $$\delta_x(E)=\begin{cases} 1, & x\in E \\ 0, & x\not\in E \\ \end{cases} $$ Let $\mu:=m+\delta_0+\delta_1$, where $m$ denotes the ...
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1 vote
1 answer
144 views

Solution Check-Finding The Radon Nikodym Derivative

I was hoping to get my solution to part $\textbf{i}$ of this qual question regarding the Radon-Nikodym derivative checked for rigor and correctness. Then I was hoping to get advice on proceeding with ...
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  • 406
2 votes
0 answers
61 views

Change of measure in multivariate Itô diffusion processes

Let $X_t$ and $Y_t$ be $d$-dimensional Itô diffusion processes that solve following SDEs, $\mathrm{d}X_t = \alpha X_t \mathrm{d}t + \Sigma \mathrm{d}B_t\,$ where $\,B_t$ is a standard brownian motion,...
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1 vote
0 answers
58 views

When conditional expectation cannot be evaluated explicitly

I'm studying probability following Shiryaev's book and I came up with the following question, which I don't seem to find a proper example anywhere. Because of the formulation of the conditional ...
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1 vote
0 answers
176 views

Example of conditional expectation from the Radon-Nikodym theorem

I'm studying probability following Shiryaev's book, and I'm trying to understand how the Radon-Nikodym theorem applies in an actual case where I can calculate the conditional expectation of a random ...
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2 votes
0 answers
37 views

Radon-Nikodým property and uncountable basis [closed]

Let $X$ be a Banach space. It is well-known that if $(x_n)_{n \geq 1}$ is a boundedly complete Schauder basis of $X$, then $X$ has the Radon-Nikodým property (indeed, $X$ is separable and is ...
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3 votes
1 answer
193 views

Radon-Nikodym for product measure

Let $(E, \mathcal{E})$ be a measurable space, $\eta:\mathcal{E}\to [0,1]$ be a probability measure defined on it and $K:E\times\mathcal{E}\to [0, 1]$ a Markov Kernel from $(E, \mathcal{E})$ onto ...
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  • 4,314
0 votes
1 answer
49 views

Expressing the likelihood function using conditional density and Radon-Nikodym

Let $X \sim P$ for some distribution $P$ and $Y \sim Q(\cdot | X)$ for some distribution $Q$. Assume that $P$ has a density $p$, with respect to some ground measure $\mu$ and $Q(\cdot|X)$ has a ...
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  • 1,440
0 votes
0 answers
54 views

Density of probability measure with respect to a $\sigma$-finite measure always exist

Let $P$ and $Q$ be probability measures on the measurable space $(\Omega, \mathcal{F})$. My lecture notes say that both $P$ and $Q$ admit densities with respect to a $\sigma$-finite measure, e.g. $\mu ...
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