Mathematics

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).

387 questions
54
votes
3 answers

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all $x \in \mathbb{R}$. More rigorously, I could…
bcf
  • 3,268
20
votes
3 answers

Radon-Nikodym derivative of product measure

For $j=1,2$, let $\nu_{j},\mu_{j}$ be $\sigma$-finite measures on $(X_{j},\mathcal{M}_{j})$ such that $\nu_{j}\ll\mu_{j}$. I want to show that $\nu_{1}\times\nu_{2}\ll\mu_{1}\times\mu_{2}$ and that…
user44532
18
votes
1 answer

Radon–Nikodym derivative and "normal" derivative

The Radon–Nikodym theorem states that, given a measurable space $(X,\Sigma)$, if a $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ is absolutely continuous with respect to a $\sigma$-finite measure $\mu$ on $(X,\Sigma)$, then there is a measurable…
Tim
  • 49,412
15
votes
1 answer

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 $\mu$ is essentially defined by a collection of…
fromscratch
  • 177
13
votes
1 answer

Show that an increasing function has derivative $0$ a.e.

Let $0<p<1$ and define $F:[0,1]\rightarrow[0,1]$ by $$F(x)=\begin{cases} pF(2x),&x\in\left[0,\frac12\right]\\ p+qF(2x-1),&x\in\left[\frac12,1\right] \end{cases}$$ where $q=1-p$ . I would like to prove that $F'(x)=0$ a.e. I am working my way through…
saulspatz
  • 54,054
12
votes
1 answer

Computing Radon-Nikodym derivative

I learned Radon-Nikodym theorem in class and I know what exactly it is. But I am not sure about how to compute Radon-Nikodym derivative... Any reference does not explicitly say about how to compute Radon-Nikodym derivative.. Can anybody help me…
Detectives
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11
votes
2 answers

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 $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \rightarrow \Bbb R.$ We can then consider the…
John D
  • 1,988
11
votes
1 answer

Distributional derivative of measures and Radon-Nikodym derivative.

The distributional derivative of a function $f$ is the function $g$ that verifies $\int f\phi'=-\int g\phi$ . In particular, if f is differentiable, the distributional derivatives is the same as the standard derivative. Now similarly, if $\mu$ is a…
edamondo
  • 1,813
9
votes
2 answers

Calculating Radon Nikodym derivative

I am trying to calculate the Radon-Nikodym derivative for $\mu = m + \delta_0$ where $m$ is Lebesgue measure over a compact subset of $\mathbb R$ and $\delta_0$ is Dirac measure at $0$. Clearly, $m \ll \mu$ and $\mu \perp \delta_0$ . Therefore, the…
Ram
  • 1,718
9
votes
1 answer

$G$ 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 exists a $G$ -invariant Radon measure $\mu$ on $X$…
Carlos Esparza
  • 3,817
9
votes
1 answer

Finding Radon-Nikodym derivative

Let $m$ be Lebesgue measure on $\mathbb R_+=(0,\infty)$ and $\mathcal A = \sigma\left(( \frac 1{n+1} , \frac 1n ]:n=1,2,...\right)$. Define a new measure $\lambda$ on $\mathcal A$, for each $E \in \mathcal A$, by $\lambda(E)= \int_E fdm $, where…
bellcircle
  • 3,101
8
votes
1 answer

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 change of measure. In the question below, however,…
Jan Stuller
  • 1,279
8
votes
2 answers

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 literature. As such I've tried to very, very slowly break down…
tisPrimeTime
  • 942
8
votes
1 answer

Radon-Nikodym-derivative as a martingale

At the beginning of all the stuff about Girsanov theorem, we introduced the Radon-Nikodym derivative as $Z_\infty := \frac{d \mathbb{Q}}{d \mathbb{P}}\vert_{\mathcal{F}_\infty}$. Next, we considered the following: \begin{align} Z_t =…
tubmaster
  • 758
8
votes
1 answer

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$ , and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is standard $\mathbb P$ -Brownian motion.…
BCLC
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