this is true for indicator functions, since ∫ 1Adπ = π(A) = ∫ A hπ,μdμ by definition of the Radon-Nikodym derivative. Then by linearity of the integral, the claim is also true for simple functions. Then we can show the claim for non-negative measurable functions, by approximating by simple functions. Indeed, if f ≥ 0 is measurable, then there exists a sequence of simple functions 0 ≤ sn ↑ f suc...

We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>\psi(x,y,t)\}. assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix $\mathbb R^{m}$) concerning $\psi$ $\Omega$...

The mean density of a random closed set Θ in R with Hausdorff dimension n is the Radon-Nikodym derivative of the expected measure E[H(Θ ∩ · )] induced by Θ with respect to the usual d-dimensional Lebesgue measure. We consider here inhomogeneous Boolean models with lower dimensional typical grain. Under general regularity assumptions on the typical grain, related to the existence of its Minkowsk...

The solution of a stochastic control problem depends on the underlying model. The actual real world model may not be known precisely and so one solves the problem for a hypothetical model, that is in general different but close to the real one; the optimal (or nearly optimal) control of the hypothetical model is then used as solution for the real problem. In this paper we assume that, what is n...

We address the question of how the approximation error/Bellman residual at each iteration of the Approximate Policy/Value Iteration algorithms influences the quality of the resulted policy. We quantify the performance loss as the Lp norm of the approximation error/Bellman residual at each iteration. Moreover, we show that the performance loss depends on the expectation of the squared Radon-Niko...

The average distance problem finds application in data parameterization, which involves “representing” the data using lower dimensional objects. From a computational point of view it is often convenient to restrict the unknown to the family of parameterized curves. However this formulation exhibits several undesirable properties. In this paper we propose an alternative variant: the average dist...

A Theorem of Lyapunov states that the range R(μ) of a non–atomic vector measure μ is compact and convex. In this paper we give a condition to detect the dimension of the extremal faces of R(μ) in terms of the Radon–Nikodym derivative of μ with respect to its total variation |μ|: namely R(μ) has an extremal face of dimension less or equal to k if and only if the set (x1, . . . , xk+1) such that ...

Given a completely positive ~CP! map T, there is a theorem of the Radon– Nikodym type @W. B. Arveson, Acta Math. 123, 141 ~1969!; V. P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 ~1986!# that completely characterizes all CP maps S such that T2S is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon–Nikodym formali...

This paper deals with some operator representations Φ of a weak*-Dirichlet algebra A, which can be extended to the Hardy spaces H(m), associated to A and to a representing measure m of A, for 1 ≤ p ≤ ∞. A characterization for the existence of an extension Φp of Φ to L(m) is given in the terms of a semispectral measure FΦ of Φ. For the case when the closure in L(m) of the kernel in A of m is a s...