arXiv subjects
Publications and source records attributed to Ondřej Kalenda.
We prove that hereditarily Lindelöf space which is $F_{σδ}$ in some compactification is absolutely $F_{σδ}$. In particular, this implies that any separable Banach space is absolutely $F_{σδ}$ when equipped with the weak topology.
We investigate convergence of martingales adapted to a given filtration of finite $σ$-algebras. To any such filtration we associate a canonical metrizable compact space $K$ such that martingales adapted to the filtration can be canonically represented on $K$. We further show that (except for trivial cases) typical martingale diverges at a comeager subset of $K$. `Typical martingale' means a martingale from a comeager set in any of the standard spaces of martingales. In particular we show that a typical $L^1$-bounded martingale of norm at most one converges almost surely to zero and has maximal possible oscillation on a comeager set.