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).
388 questions
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Proof of a conditional expectation property : $\mathbb{E}(gf)=g\mathbb{E}f$
I need to prove a property about the linear operator defined by the following theorem:
Let $(X,\mathcal{B},\mu)$ be a finite measure space and $\mathcal{A}\subset \mathcal{B}$ a sub $\sigma-$algebra. ...
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Fourier Transform of mutually singular measures
The Fourier Transform $\int_{\mathbb R^d}e^{-2\pi is\cdot x}\mathrm{d}\mu$ is an injective linear map from the space of bounded-variation Borel measures to the space of continuous functions; so in ...
- 1,664
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How to show that the Radon-Nikodym density is $M_{t\wedge T}$?
Let $(\Omega, (\mathcal{F}_t)_{0\leq t\leq 1}, P)$ be some filtered probability space where the filtration satisfiy the usual assumptions. This means $\mathcal{F}_0$ contains all null sets of $\...
- 2,045
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Levy measures and symmetry
I am struggling with a measure theory problem that seems simple, but my intuition tells me I am making a mistake somewhere. I would appreciate your help. Here, all measures are Levy measures.
Let's ...
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pdf at points where cdf has kinks
I know that a pdf (probability density function) of a random variable $X$ is the Radon-Nikodym derivative of the distribution of $X$ (which is a pushforward measure of the underlying probability ...
- 3,719
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Application of Lebesgue Decomposition Theorem
I am trying to solve the following exercise: Let $m$ denote the Lebesgue measure , and let $\lambda$ be a real valued positive measure. Argue that there is a set $A$ of full Lebesgue measure (i.e. $m(...
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Radon-Nikodym derivative along a Markov chain
Let $(X,\mathcal{F})$ be a measurable space. $\nu,\mu$ are two $\sigma$-finite measure on $\Omega$ such that $\mu\ll \nu$. $P:X\times \mathcal{F}$ is a Markov kernel.
One can shows $\mu P\ll \nu P$ as ...
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Absolute continuity almost everywhere and Radon Nikodym derivative
Let $(\mathcal{X}, \mathcal{F})$ and $(\mathcal{Y}, \mathcal{G})$ be measure spaces.
Let $\mu \ll \nu$ measures on $\mathcal{X}$ and $K:\mathcal{X}\times\mathcal{G}\rightarrow[0,1]$ a Markov kernel. ...
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Absolute continuity of markov kernel
Let $(\mathcal{X}, \mathcal{F})$ and $(\mathcal{Y}, \mathcal{G})$ be measure spaces and let $K:\mathcal{X}\times\mathcal{G}\rightarrow [0,1]$ be a markov kernel connecting these two spaces.
Let $\mu$ ...
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Markov kernel applied to Radon-Nikodym derivative
Let $(\mathcal{X}, \mathcal{F})$ and $(\mathcal{Y}, \mathcal{G})$ be measure spaces and $K:\mathcal{X}\times\mathcal{G}\rightarrow [0,1]$ be a Markov kernel. Let $\mu \ll \nu$ be $\sigma$-finite ...
- 956
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Complex measures. Radon-Nikodym derivative [closed]
Let $S$ be a compact Haursdorff space. Let $\mu$ be a regular complex Borel measure on S. Let $f$ be a non-negative real function on S. Is it the same to define $\sigma$ by $$d\sigma=fd\mu$$ and by$$\...
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Bounds for infinity norm and Radon-Nikodym derivative under certain given conditions.
I have been stuck for a while with the following couple of exercises:
(1) Let $(X, \mathcal{M}, \mu )$ be a measure space, $(f_n)_{n \in \mathbb{N}}$ a sequence of functions in $L^{1}(X, \mathcal{M}, \...
- 607
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Definition and motivation for Radon-Nikodym property
The following definition is from Kalton's book on Banach space theory.
I have a question regarding the definition; why is the space specifically $L^1([0,1])$. I know there is another definition ...
- 3,389
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What does conditional expectation mean for continuous random variables?
The Borel-Kolmogorov Paradox shows trying to condition on a measure zero event like $Y = y$ can give you different answers if you describe the event $Y = y$ using different random variables. So not ...
- 8,062
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Existence of a Measurable Subset with Uniform Lower Bound for the Ratio of Two Finite Measures
The Problem:
Let $\mu$ and $\nu$ be measures on a measurable space $(X, \mathcal{F})$ such that $\mu$ is $\sigma$-finite, $\nu$ is nonzero (i.e., $\nu(X) > 0$) and finite. Assume also that $\nu \ll ...
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