We present a general way of defining various reduction games on ω which “represent” corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ-measurable functions. These games turn out to be useful as a combinatorial tool for the...

We show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property. The goal of this paper is to show yet another example of nonduality between measure and category. Suppose that F is a nonprincipal filter on ω. Identify F ...

in proposition 2.6 in (g‎. ‎gruenhage‎, ‎a‎. ‎lutzer‎, ‎baire and volterra spaces‎, ‎textit{proc‎. ‎amer‎. ‎math‎. ‎soc.} {128} (2000)‎, ‎no‎. ‎10‎, ‎3115--3124) a condition that‎ ‎every point of $d$ is $g_delta$ in $x$ was overlooked‎. ‎so we‎ ‎proved some conditions by which a baire space is equivalent to a‎ ‎volterra space‎. ‎in this note we show that if $x$ is a‎ ‎monotonically normal $t_1$...

Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, generalisation of Lebesgue integral. We study axiomatic strength for various classes functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. prove that, over $\mathsf{RCA}_0$: (i) continuous functions equivalent to $\mathsf{WKL}_0$; (ii) Baire class 1 $\mathsf{AC...