Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

ℒ denotes the Lebesgue measurable subsets of ℝ and $ℒ_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ\$ℒ_0$ has a perfect subset Q ∈ ℒ\$ℒ_0$ which is a subset of or misses M (a similar statement omitting "is a subset of or" characterizes $ℒ_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the "Marczewski measurable sets" and the σ-ideal $(s^0)$ which we call the "Marczewski null sets". M ∈ (s) if every perfect set P has a perfect subset Q which is a subset of or misses M. M ∈ $(s^0)$ if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of $G_δ$ sets which can be used to give similar "Marczewski-Burstin-like" characterizations of the collections $B_w$ (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of $F_σ$ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of $B_r$ (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and $U_0$ (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed

Convergence in distribution of random compact sets in Polish spaces

Let [phi],[phi]1,[phi]2,... be a sequence of random compact sets on a complete and separable metric space (S,d). We assume that P{[phi]n[intersection]B=[empty set]}-->P{[phi][intersection]B=[empty set]} for all B in some suitable class and show that this assumption determines if the sequence {[phi]n} converges in distribution to [phi]. This is an extension to general Polish spaces of the weak convergence theory for random closed sets on locally compact Polish spaces found in Norberg [1984. Convergence and existence of random set distributions. Ann. Probab. 12, 726-732.]Convergence in distribution Random compact sets Polish spaces

A note on the multivariate local time intensity of exchangeable interval partitions

Consider an exchangeable interval partition [Xi] of [0,1] with associated local time [xi]. We give a precise description of the set where the intensity measure E[xi]n, n[greater-or-equal, slanted]2, admits a continuous density. We also examine to what extent that set exhausts the continuity points of the density.Exchangeable interval partitions Local time