Continuous approximation of random function

AbstractModifying the methods of Lee [J. Math. Anal. Appl. 61 (1977), 1–6], we show that each μ-measurable mapping f on a normal space T into a separable linear metric space E is almost continuous, where μ is a Radon probability measure. It is shown that for every ε > 0 there exists a compact subset Kε ⊂ T with μ(Kε) > 1 − ε and an elementary function g(t) = ∑ni = 1 hi(t) xi such that μ(t ϵ Kε; f(t) ≠ g(t)) < ε, where xi ϵ E and hi(t) are real bounded continuous functions with disjoint supports

Nuclear subspace of L0 and the Kernel of a linear measure

AbstractLet E be a locally convex space. Then E is nuclear metrizable if and only if there exists a σ-additive measure μ on E′ such that L: E → L0(E′, μ), L(x) = 〈x, ·〉, is an isomorphism. Let E be quasi-complete or barrelled. Suppose that there exists a σ-additive measure ν on E satisfying (E′, τν)′ ⊃ E. Then E′b is an isomorphic subspace of L0(E, ν) and nuclear, where b is the strong dual topology and τν is the L0(E, ν) topology. In the case where E is an LF space, for a random linear functional L: E → L0(Ω, U, P), the next conditions are equivalent: (a) The cylinder set measure μ on E′ determined by L is σ-additive and (b) xn → 0 in E implies that L(xn) → 0, P-a.s

Orbital Decay and Tidal Disruption of a Star Cluster: Analytical Calculation

The orbital decay and tidal disruption of a star cluster in a galaxy is studied in an analytical manner. Owing to dynamical friction, the star cluster spirals in toward the center of the galaxy. Simultaneously, the galactic tidal field strips stars from the outskirts of the star cluster. Under an assumption that the star cluster undergoes a self-similar evolution, we obtain the condition and timescale for the star cluster to reach the galaxy center before its disruption. The result is used to discuss the fate of so-called intermediate-mass black holes with >10^3 M(sun) found recently in young star clusters of starburst galaxies and also the mass function of globular clusters in galaxies.Comment: 12 pages, 1 PS file for 2 figures, to appear in The Astrophysical Journa