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).
335
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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 ...
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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 ...
- 357
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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 ...
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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 ...
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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 $\mu(A) = \lambda(A^3)$. ...
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42
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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 $x\in X$, ...
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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 ...
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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 ...
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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 $(\mathsf{X}, \mathcal{X})$ ...
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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{...
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How to determine whether a measure space is $\sigma$ finite
Question: Let $\lambda, \mu, \nu$ be measures on a measurable space $(X,M)$. Suppose $\nu \ll \mu$ and $f:X \to [0, \infty]$ is measurable. Show that $$\int f\mathrm d\nu= f\frac{\mathrm d \nu} {\...
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Is the Converse of the Radon-Nikodym Theorem true?
I'm curious as to whether the converse of the Radon-Nikodym theorem holds:
Converse Theorem: let $\mu.\nu$ be measures on $(\Omega,\Sigma)$ such that
$$\nu : A\mapsto \int_A f \ d\mu$$
for some ...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 $...
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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 $\...
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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 ...
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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 ...
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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{...
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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 ...
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Radon-Nikodym derivative in a compact Hausdorff space
Let $X$ be a compact Hausdorff space, $m$ be a regular non-atomic probability measure defined on the Borel $\sigma$-algebra of $X$, and $g$ an autohomeomorphism of $X$. Suppose that the image measure $...
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How are these statements of the Radon-Nikodym theorem the same
I am reading An Informal Introduction to Stochastic Calculus with Applications and I've come across this statement of the Radon-Nikodym theorem.
"Consider the probability space $(\Omega ,\mathcal ...
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Applying Radon-Nikodym Theorem to Rewrite a Path Integral
I'm working on a problem involving a path integral of the form:
$$N = \int X(\omega) dP(\omega).$$
Here, $\omega$ represents sawtooth-shaped paths going from time $0$ to $t$, and each tooth of the ...
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Sigma-algebra used in the theorem of Lebesgue-Radon-Nikodym of Rudin's Real and Complex Analysis
The theorem of Lebegue-Radon-Nikodym in page 121 of Rudin's Real and Complex Analysis reads as:
Let $\mu$ be a positive $\sigma$-finite measure on a $\sigma$-algebra $\mathfrak{M}$ on a set $X$, and ...
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Correspondence between set of proba measure with square-integrable density and set of adapted processes
Let W be a Brownian motion, T a finite time horizon and $(\Omega, \Bbb{P}, \Bbb{F})$ a probability space with $\Bbb{F} = \Bbb{F}^W$ the natural augmented filtration of $W$. For any $\Bbb{F}$-adapted ...
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Show that $g \in C([a,b]) \cap BV([a,b])$ with $\mu(E) = 0 \implies \mu(g(E)) = 0$ is in $AC([a,b])$
Let $g : [0,1] \to \mathbb{R}$ be continuos and of bounded variation such that has the (N) proprety ( i.e. $\mu(E) = 0 \implies \mu(g(E)) = 0$ ), show that $g$ is absolutely continuos.
My attempt :
To ...
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Question about proof of factorization theorem for sufficient statistics (Shao Theorem 2.2)
The setup in Shao's Theorem 2.2 (Factorization theorem) is as follows:
Let $X$ be a sample from $P\in \mathcal P$, a family of probability measures on $(\mathbb R^n,\mathcal B(\mathbb R^n))$ dominated ...
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Understanding Radon Nikodym proof in the book Measure Theory from Donald L.Cohn
I'm reading the book Measure Theory (second edition) from Donald L.Cohn and
have a problem in understanding his proof of the $\sigma$-finite case of his
proof of the Radon-Nikodym theorem. In the book ...
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Question regarding Lebesgue decomposition of a finite measure
Let $P$ and $Q$ denote finite measures. Let $\mu$ denote a finite measure that dominates them both, and $p, q$ the Radon-Nikodym derivatives of $P, Q$, respectively, with respect to $\mu$.
The ...
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Notation of the Radon-Nikodym derivative [duplicate]
A question regarding notation of the radon nikodym derivative:
If the derivative $\frac{d\nu}{d\mu}:=f$ exists, some authors write something like $d\nu = f d\mu$, i.e., they treat the derivative as if ...
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Writing Radon Nikodym derivative of joint distributions in terms of marginals and conditionals
Given two random variables $X, Y$ and two joint probability distributions $P_{X, Y}$ and $Q_{X, Y}$ on the same product space $(\mathcal{X}\times\mathcal{Y}, \mathcal{A}\times\mathcal{B})$ that are ...
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Probability conditioning on $X= Y$
Given a random variable $X\sim \text{Exp}(1)$, I would like to condition on the event that $$``\text{$X$ behaves like an $\text{Exp}(1/2)$ random variable}".$$
Is there a way to define this event and ...
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Codomain of almost everywhere defined functions
So I am currently wondering about how one can write down the codomain of a function that is only almost everywhere defined.
In particular, I am looking at the radon nikodym derivative $f$, which is ...
- 357
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absolutely continuous ($\sigma$-) finite measures
I got two questions:
If I have two measures $\nu$ and $\mu$ with $\nu << \mu$ and we know that $\nu$ is (i) finite (e.g. a probability measure) or (ii) $\sigma$-finite. Can we draw any ...
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What is the codomain of the Radon–Nikodym derivative and why$
My question is regarding the Radon–Nikodym derivative $\frac{d\nu}{d\mu}$, when $\nu \ll \mu$ and both measures are $\sigma$-finite. So Wikipedia says in the article about the Radon–Nikodym theorem ...
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$\sigma$-linearity of the Radon-Nikodym operator.
Let $(X,\mathcal X,\nu)$ be a probability space, let $\{ \mu_n \}_{n\in\mathbb N}$ be a set of finite positive measures on $(X,\mathcal X)$ that are all absolutely continuous with respect to $\nu$. I ...
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Radon-Nikodym Derivative leading to a conditional measure
Let $\mu(\cdot \mid y)$ be a conditional measure for all $y \in \mathcal{A}$ and $\mu_0$ be another measure. These two measures are defined on the same measurable space $(\mathcal{A}, \sigma(\mathcal{...
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Sequence of ratios of measures with bounded Radon-Nikodym derivative cannot diverge
Assumptions: Let $\mu$ and $\nu$ be positive and finite measures on $(X, \Sigma)$. Suppose that $\mu$ is absolutely continuous with respect to $\nu$, so that the Radon-Nikodym derivative of $\mu$ with ...
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Conditional expectation Folland 3.2.17
Let $(X,M,\mu)$ be a finite measure space and $N$, a sub-$\sigma$-algebra of $M$. And $\nu=\mu|_N$. If $f\in L^1(\mu)$, there exists $ g\in L^1(\nu )$, such that $\int_{E}fd\mu = \int_{E}gd\nu $ for ...
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Applying Radon-Nikodym and Riesz Representation Theorem to Prove a Proposition
I have a paper that I am working on. I wanted to know can a function $\nu$ which is absolutely continuous with respect to measure $\mu$ be represented as an integral where the function is having a ...
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Relationship between the Radon Nikodym derivative with respect to two measures in both directions
Background
The Radon-Nikodym Theorem tells us that for two sigma finite measures $\mu$ and $\nu$, that $\nu$ is absolutely continuous with respect to $\mu$ if and only if there exists a function $f$ ...
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Question on Radon-Nikodym derivative: Showing that $\int_Afd\mu_1=\int_Afg_{\mu_2}(\mu_1)d\mu_2$ for every Borel set $A$ when $\mu_1,\mu_2$ are finite
Let $\mu_1,\mu_2$ be two finite Radon measures on $\mathbb{R}^n$ such that $\mu_1$ is absolutely continuous w.r.t. $\mu_2$, i.e. $\mu_1\ll\mu_2$, and denote by $g_{\mu_2}(\mu_1)$ the Radon-Nikodym ...
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Product measurability of the Radon-Nikodym derivative of two Marcov kernels
Let $(\mathcal{X},\mathcal{A})$ and $(\mathcal{Y},\mathcal{B})$ are Borel spaces.
Let $\kappa_i: \mathcal{B} \times \mathcal{X} \to [0,1], ~i=1,2$ be Marcov kernels from $(\mathcal{X},\mathcal{A})$ to ...
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Computation of a limit using a corollary of Lebesgue's differentiation theorem
Let $\nu$ and $\mu$ regular measures on $\mathbb{R}$ absolutely continuous with respect to the lebesgue measure $\lambda$.
I am asked to compute the following limit:
$$
\lim_{r \to 0} \frac{(\mu \...
- 536
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0
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Refined Lebesgue Decomposition: $\nu_d \le \nu_s$ and $\nu_c \ge \nu_a$
In my lecture on measure theory, we wanted to derive a refined decomposition of the Lebesgue decomposition.
Let $\nu$ be a locally finite measure on $(\mathbb R, \mathcal B(\mathbb R))$. I already ...
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