We study the σ-ideal of Haar null sets on Polish groups. It is shown that on a non-locally compact Polish group with an invariant metric this σ-ideal is closely related, in a precise sense, to the σ-ideal of non-dominating subsets of ${\omega }^{\omega }$. Among other consequences, this result implies that the family of closed Haar null sets on a Polish group with an invariant metric is Borel in the Effros Borel structure if, and only if, the group is locally compact. This answers a question of Kechris. We also obtain...

The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected...

The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected...

Let X be a Polish space, and let ${\left(A_p\right)}_{p\in \omega }$ be a sequence of $G_{\delta }$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether ${\cup }_{p\in \omega }{A}_p$ is a true ${\sum }_3^0$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true ${\sum }_3^0$.