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...

in this work are studied the jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

A method is proposed for detecting whether two CCITT group 4 images were scanned from the same document. Features are extracted from rectangular patches of text and compared with a modified Hausdorff distance measure. Two images are said to be ‘‘equivalent’’ (i.e., they were scanned from the same document) if the Hausdorff measure finds that a specified number of features are located within a g...

For a closed subset K of a compact metric space A possessing an α-regular measure μ with μ(K) > 0, we prove that whenever s > α, any sequence of weighted minimal Riesz s-energy configurations ωN = {x i,N} N i=1 on K (for ‘nice’ weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore, if K is an α-rectifiable com...