Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and stron...

Resource-bounded measure is a generalization of classical Lebesgue measure that is useful in computational complexity. The central parameter of resource-bounded measure is the resource bound ∆, which is a class of functions. When ∆ is unrestricted, i.e., contains all functions with the specified domains and codomains, resource-bounded measure coincides with classical Lebesgue measure. On the ot...

Abstract We prove a version of the Lebesgue differentiation theorem for mappings that are defined on measure space and take values into metric space, with respect to basis induced by von Neumann lifting. As consequence, we obtain lifting sections measurable Banach bundle disintegration vector measures whose target is Radon–Nikodým property.

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which purely singular with respect to Lebesgue measure. Can such leave an infinite, $\sigma$-finite absolutely continuous measure? In this paper, using old criterion due Katznelson, we show the answer question no.

This paper is mainly concerned with scrambled sets for Luroth map. It shown that all have null Lebesgue measure and there exists a set full Hausdorff dimension.