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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$\textbf{Theorem 4.3}$ 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 ...
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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 ...
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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 ...
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### 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 $k\in M$,...
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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 ...
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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 ...
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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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### 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 ...