The authors consider the problem of searching for activation in brain images obtained from functional magnetic resonance imaging and the corresponding functional signal detection problem. They develop a Bayesian procedure to detect signals existing within noisy images when the image is modeled as a scale space random field. Their procedure is based on the Radon–Nikodym derivative, which is used...

Idempotent integration is an analogue of the Lebesgue integration where σ-additive measures are replaced by σ-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon–Nikodym derivatives, which turns out to be crucial in all of these applications, was prov...

A Banach spaceX with the property that every absolutely continuousX−valued function is almost everywhere differentiable is said to be a Radon-Nikodym space [7, pp. 217–219] or [2, 13] (see also [3]). For example, every reflexive Banach space (in particular, every Hilbert space) is a Radon-Nikodym space, but the space L∞ [0, 1] of all K−valued, essentially bounded functions defined on the interv...

Let F denote a finite field of q elements. Define a Nikodym set in Fn to be a set B ⊂ Fn such that for each x ∈ Bc, there exists a line L such that L ∩ Bc = {x}. In this note we show that the size of every Nikodym set is at least Cn · q n, where Cn depends only on n.

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C (and C on every curve)...

We consider two Gaussian measures. In the "initial" measure the state variable is Gaussian, with zero drift, and time-varying volatility. In the "target measure" the state variable follows an Ornstein-Uhlenbeck process, with a free set of parameters, namely the time-varying speed of mean-reversion. We look for the speed of mean reversion that minimizes the variance of the Radon-Nikodym derivati...

We design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares Monte Carlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates with respect to the unknown change of measu...

Let X be a Banach space and (Ω,Σ, λ) be a finite measure space, 1 ≤ p < ∞. It is shown that L(λ,X) has the Near Radon-Nikodym property if and only if X has it. Similarly if E is a Köthe function space that does not contain a copy of c0, then E(X) has the Near Radon-Nikodym property if and only if X does.

Let ~ and m be excessive measures for a right Markov process X and let Q~ and Qm be the associated stationary Kuznetsov processes. We show that if ~ and m are harmonic, then Q~ ~ Qm if and only if ~ is quasi-bounded by m in the sense that ~ = where each term in the sum is an excessive measure dominated by m. This result allows us to describe the Lebesgue decomposition of Q~ relative to Qm and t...