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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### Can we show $\frac{\nu\left(B_\varepsilon(x)\right)}{\mu\left(B_\varepsilon(x)\right)}\xrightarrow{\varepsilon\to0+}\frac{{\rm d}\nu}{{\rm d}\mu}(x)$?

Let $(E,\mathcal E,\mu)$ be a finite measure space and $\nu$ be a finite signed measure on $(E,\mathcal E)$ with $\nu\ll\mu$. By the Radon-Nikodým theorem, $$\nu=f\mu\tag1$$ for some $f\in L^1(\mu)$. ...
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### Extending proof of positive measure to real measure for Radon Nikodym

The Wikipedia proof of Radon-Nikodym first shows for positive measures, then extends to real measures by applying the positive measure Radon-Nikodym theorem to the unique + and - (positive) measures ...
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### Constructing a specific measure from a given measure such that its Rdon Nikodym derivative takes specific values

Take a measure space $(\Omega, \mathcal{A}, \tau, P)$, with $P$ being a probability measure, and $\tau$ being a topology on $\Omega$. I would like to construct a measure $Q$ such that $Q$ is ...
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### Integrating a measure with respect to itself

Take a measure space $(\Omega, \mathcal{A}, P)$, with $P$ being a probability measure. I want to construct a second measure from $P$ via $Q(E) := \int_E P(\omega) \; dP(\omega)$. In other words, I ...
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### Radon Nikodym derivative of a restriction of a measure.

So how I see it $\nu <<\mu$ on $\mathcal{N}$. Thus there exists a Borel measurable function such that $\nu(A)=\int_Ahd\mu|\mathcal{N}=\int_Ahd\mu$ where the equality holds by the observations ...
### If $\lambda$ is a signed measure and $\lambda\ll\mu$, then $\lambda^{+}\ll\mu$ and $\lambda^{-}\ll\mu$
Original: Let $\mu$ be a finite measure on $\mathbb{R}^k$ dominated by the Lebesgue measure. Does $\mu$ need to be a Radon measure? Updated Question: Does anyone have a reference to the fact that if ...