The theory of Hausdorff dimension provides a general notion of the size of a set in a metric space. We define Hausdorff measure and dimension, enumerate some techniques for computing Hausdorff dimension, and provide applications to self-similar sets and Brownian motion. Our approach follows that of Stein [4] and Peres [3].

Modifying the Hausdorff-Buseman metric, we obtain a compatible metric with the Fell-Matheron topology on the space of closed subsets of a locally compact Hausdorff second countable space. We also give an alternative expression of this metric, and two more compatible metrics, a metric of summation form and a modified Rockafellar-Wets metric. Through the study of the relation between the original...

Comparing face images using the Hausdorff distance is one of the face matching and fast screening techniques. As a fast screening technique, the computational efficiency is a key issue. An efficient method for Hausdorff distance-based face matching and fast screening is proposed. The method utilises dominant points, instead of edge maps, as features for measuring similarity. A new formulation o...

We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Fréchet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quo...

Here, instead of taking a σ-algebra as given and then defining a measure on this σ-algebra (namely, on the measurable sets), we take an outer measure as given and then define measurable sets using this outer measure. Carathéodory’s theorem states that the collection M of ν-measurable sets is a σ-algebra and that the restriction of ν to M is a complete measure. Suppose that (X, ρ) is a metric sp...

‎It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$‎. ‎Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology‎, ‎which is compact by the Banach--Alaoglu theorem‎. ‎We prove that the compact Hausdorff space $X$ can ...

The aim of this paper is to present a fuzzy counterpart method of constructing the Hausdorff quasi-uniformity of a crisp quasi-uniformity. This process, based on previous works due to Morsi cite{Morsi94} and Georgescu cite{Georgescu08}, allows to extend probabilistic and Hutton $[0,1]$-quasi-uniformities on a set $X$ to its power set.  In this way, we obtain an endofunctor for each one of the c...