We provide self-contained proof of a theorem relating probabilistic coherence of forecasts to their non-domination by rival forecasts with respect to any proper scoring rule. The theorem recapitulates insights achieved by other investigators, and clarifi es the connection of coherence and proper scoring rules to Bregman divergence.

We consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function $u \colon T \to \mathbb {R}$ such that $f(x) = \limsup _{t \to \infty } u(x_{0},\dots,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

Rosłanowski and Shelah [Small-large subgroups of the reals, Math. Slov. 68(3) (2018) 473–484] asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with [Formula: see text] that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper, we give affirmative answers for the general case.

Rosłanowski and Shelah [Small-large subgroups of the reals, Math. Slov. 68 473–484] asked whether every locally compact non-discrete group has a null but non-meager subgroup, and converse...

We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of ZFC \ CH. In this paper we present a sufficient condition to be consistently the set of cardinalities of branches of Kurepa trees.

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