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).

258 questions
Filter by
Sorted by
Tagged with
1 vote
66 views

### Converse to Lebesgue Differentiation Theorem

Suppose we are given two finite positive Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$ such that the function $$f(x) := \limsup_{r\to 0} \frac{\mu(B(x,r))}{\nu(B(x,r))}$$ is in $L^1(\nu)$. Is it ...
1 vote
27 views

• 705
39 views

1 vote
32 views

### Question on relation between radon-nikodym derivatives and the total variation distance

Let $(\mathcal{X},\mathcal{A})$ be a measure space on which we have defined two probability measures $P$ and $Q$. I am reading some notes online which makes the following jump without explanation in ...
64 views

### Radon-Nikodym derivative of pushforward measures and Girsanov theorem

Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
• 571
22 views

### Is the Radon-Nikodyn derivative unique (under a few assumptions)

For two measures $\mu$ and $\nu$ (with $\mu \ll \nu$) over a topological space $X$, $\mu$ can be expressed as $$\mu(A)= \int_A \frac{d\mu}{d\nu}d\nu$$ and the function $\frac{d\mu}{d\nu}$ is the Radon-...
• 352
1 vote
51 views

1 vote
101 views

• 587
15 views

• 1,785
1 vote
105 views

### Folland Theorem 3.22

At the start of the proof for Theorem 3.22, Folland says that $dv=d\lambda + fdm$ implies $d|v|=d|\lambda| + |f|dm$. I get why this is the case for positive and signed measures, but I'm not sure how ...
63 views

• 958
73 views

### Deduce that $Z_m$ is the Radon-Nikodym-density $Z_m=\frac{dQ_m}{dP}$ of the probability measure $Q_m$, which is equivalent to P

The information given: Consider an arbitrage-free one-period financial market model $(S^0;S)$. We have a risk free asset $S^0$ with $S_0^0=1,S_1^0=1$ so the risk free rate $r=1$. We have a risky asset ...
• 83
95 views

Problem statement: For a measure space $(\Omega, \mathcal{F}, \mathrm{μ})$ : ${μ}$ is positve and finite, show the following: The set function $\nu(A) = \int_Af d\mu$ , $A \in \mathcal{F}$ defines ...
• 13
54 views

### How to calculate the Doléans-Dade exponential/Radon–Nikodym density given an SDE of a defaultable bond?

Roughly speaking, when we have the dynamics of a stock process be given by the SDE: $$dS_t = \mu_tS_tdt + \sigma_tS_tdW_t,$$ and if we can choose a cash account $B_t = e^{\int_0^t r_s ds}$ as ...
43 views

### Show that $\frac{dP}{d\nu} = \sum_{n=1}^{\infty} a_n \frac{d P_n}{d \nu}$?

Let $P=\sum_{n=1}^{\infty} a_n P_n$ be a probability measure where $a_n >0$, $\{P_n\}$ be a sequence of probability measures and $\sum_{n=1}^{\infty} a_n=1$. If $P \ll \nu$ where $\nu$ is a sigma-...
42 views

### Density w.r.t. measure on the unit simplex

Let $M$ be a measure on the unit simplex $S_d=\{ \mathbf{x} \in \mathbb{R}_+^{d}: ||\mathbf{x}||_1=1 \}$ where $||.||_1$ is the $\mathcal{L}_1$ norm and $\mathbb{R}^d_+=[0, \infty)^{d}$. A text I am ...
• 397
337 views

### 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 ...
• 822
54 views

### delta function as density w.r.t. counting measure

As we know, the delta function is not a Radon-Nikodym density with respect to the Lebesgue measure. If we choose the counting measure $\mu$, which assigns to every set the number of its elements, then ...
• 57
1 vote
28 views

### Change in expectation due to Bayesian update

Consider a prior $P$ on some finite set $X$, and a posterior $Q$ on $X$ formed via a Bayesian update of $P$ given some data $y$. I am interested in finding alternative ways to write the following ...
• 323
67 views

### To show absolute continuity of measures.

Let $\delta_x$ denote the measure defined by $$\delta_x(E)=\begin{cases} 1, & x\in E \\ 0, & x\not\in E \\ \end{cases}$$ Let $\mu:=m+\delta_0+\delta_1$, where $m$ denotes the ...
1 vote
144 views

### Solution Check-Finding The Radon Nikodym Derivative

I was hoping to get my solution to part $\textbf{i}$ of this qual question regarding the Radon-Nikodym derivative checked for rigor and correctness. Then I was hoping to get advice on proceeding with ...
• 406
61 views

### Change of measure in multivariate Itô diffusion processes

Let $X_t$ and $Y_t$ be $d$-dimensional Itô diffusion processes that solve following SDEs, $\mathrm{d}X_t = \alpha X_t \mathrm{d}t + \Sigma \mathrm{d}B_t\,$ where $\,B_t$ is a standard brownian motion,...
• 31
1 vote
58 views

### When conditional expectation cannot be evaluated explicitly

I'm studying probability following Shiryaev's book and I came up with the following question, which I don't seem to find a proper example anywhere. Because of the formulation of the conditional ...
1 vote
176 views

### Example of conditional expectation from the Radon-Nikodym theorem

I'm studying probability following Shiryaev's book, and I'm trying to understand how the Radon-Nikodym theorem applies in an actual case where I can calculate the conditional expectation of a random ...
37 views

### Radon-Nikodým property and uncountable basis [closed]

Let $X$ be a Banach space. It is well-known that if $(x_n)_{n \geq 1}$ is a boundedly complete Schauder basis of $X$, then $X$ has the Radon-Nikodým property (indeed, $X$ is separable and is ...
193 views

Let $(E, \mathcal{E})$ be a measurable space, $\eta:\mathcal{E}\to [0,1]$ be a probability measure defined on it and $K:E\times\mathcal{E}\to [0, 1]$ a Markov Kernel from $(E, \mathcal{E})$ onto ...
• 4,314
49 views

### Expressing the likelihood function using conditional density and Radon-Nikodym

Let $X \sim P$ for some distribution $P$ and $Y \sim Q(\cdot | X)$ for some distribution $Q$. Assume that $P$ has a density $p$, with respect to some ground measure $\mu$ and $Q(\cdot|X)$ has a ...
• 1,440
### Density of probability measure with respect to a $\sigma$-finite measure always exist
Let $P$ and $Q$ be probability measures on the measurable space $(\Omega, \mathcal{F})$. My lecture notes say that both $P$ and $Q$ admit densities with respect to a $\sigma$-finite measure, e.g. \$\mu ...