If ZFC is consistent, then each of the following are consistent with ZFC + 20 = א2: 1. X ⊆ IR is of strong measure zero iff |X| ≤ א1 + there is a generalized Sierpinski set. 2. The union of א1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size א2. 1 The authors thank the Israel Foundation for Basic Research, Israel Academy of Science. 2 Publi...

For a subshift of nite type and a xed HH older continuous function, the zero measure invariant set of points where the Birkhoo averages do not exist is either empty or carries full Hausdorr dimension. Similar statements hold for conformal repellers and two-dimensional horseshoes, and the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoo averages do n...

This derivation of (2) is not valid, because this duality applies to inferences and theorems only, not to definitions. In fact, (1) should be treated as an axiom. Accordingly, the duality of Lemma 3.3.1 would not be valid until the dual (2) of (1) is proved independently, as we do below. Our new proof will make use of Proposition 3.3.3, the original proof of which is based on duality. So, we mu...

Answering a recent question posed by Gregori, Morillas and Sapena (“On a class of completable fuzzy metric spaces”, Fuzzy Sets and Systems, 161 (2010), 2193–2205) we present two examples of non strong fuzzy metrics (in the sense of George and Veeramani).

The class of Borel sets in a locally compact space is the cr-ring generated by the compact sets [3, p. 219], while the class of weakly Borel sets is the (r-algebra generated by the closed sets [2]. The object of this note is to show that any measure defined on the class of Borel sets may be extended to the class of weakly Borel sets in a simple and canonical way (Theorem 1). There is a useful a...

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a “good” additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are consid...

We provide an extension of the notion of chain-valued frame introduced by Pultr and Rodabaugh in [Category theoretic aspects of chain-valued frames: parts I and II, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558] by relaxing the assumption that L be a complete chain. As a result of this investigation we formulate the category L-Frm of L-frames under the weaker assumption that L is a comp...