Hausdorff Density
In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point. Definition Let \mu be a Radon measure and a\in\mathbb^ some point in Euclidean space. The ''s''-dimensional upper and lower Hausdorff densities are defined to be, respectively, : \Theta^(\mu,a)=\limsup_\frac and : \Theta_^(\mu,a)=\liminf_\frac where B_(a) is the ball of radius ''r'' > 0 centered at ''a''. Clearly, \Theta_^(\mu,a)\leq \Theta^(\mu,a) for all a\in\mathbb^. In the event that the two are equal, we call their common value the s-density of \mu at ''a'' and denote it \Theta^(\mu,a). Marstrand's theorem The following theorem states that the times when the ''s''-density exists are rather seldom. : Marstrand's theorem: Let \mu be a Radon measure on \mathbb^. Suppose that the ''s''-density \Theta^(\mu,a) exists and is positive and finite for ''a'' in a set of positive \mu measure. Then ''s'' is an integer. Preiss' theorem In 1987 David Prei ...
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Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ...
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