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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 Pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon Measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures. Motivation A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \operatorname B^n or \ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Preiss
David Preiss FRS (born January 21, 1947) is a Czech and British mathematician, specializing in mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m .... He is a professor of mathematics at the University of Warwick Preiss is a recipient of the Ostrowski Prize (2011) and the winner of the 2008 London Mathematical Society Pólya Prize (LMS), Pólya Prize for his 1987 result on ''Geometry of Measures'', where he solved the remaining problem in the geometric theoretic structure of sets and measure (mathematics), measures in Euclidean space. He was an invited speaker at the ICM 1990 in Kyoto. He is a Fellow of the Royal Society (2004) and a Foreign Fellow of the Learned Society of the Czech Republic (2003). He is associate editor of the mathematical journal ''Real An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectifiable Set
In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory. Definition A Borel subset E of Euclidean space \mathbb^n is said to be m-rectifiable set if E is of Hausdorff dimension m, and there exist a countable collection \ of continuously differentiable maps :f_i:\mathbb^m \to \mathbb^n such that the m-Hausdorff measure \mathcal^m of :E\setminus \bigcup_^\infty f_i\left(\mathbb^m\right) is zero. The backslash here denotes the set difference. Equivalently, the f_i may be taken to be Lipschitz continuous without altering the definition. Other authors have different defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (measure Theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure ''μ'' on a measurable topological space (''X'', Borel(''X'')) is a precise notion of where in the space ''X'' the measure "lives". It is defined to be the largest ( closed) subset of ''X'' for which every open neighbourhood of every point of the set has positive measure. Motivation A (non-negative) measure \mu on a measurable space (X, \Sigma) is really a function \mu : \Sigma \to , +\infty. Therefore, in terms of the usual definition of support, the support of \mu is a subset of the σ-algebra \Sigma : :\operatorname (\mu) := \overline, where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on \Sigma . What we really want to know is where in the space X the measure \mu is non-zero. Consider two examples: # Lebesgue measure \lambda on the real line \mathbb . It seems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : ''absolutely continuous'' ⊆ ''uniformly continuous'' = ''cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in ,∞to each set in \R^n or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in \R^n is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is not necessarily an integer. These measures are fundamenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pertti Mattila
Pertti Esko Juhani Mattila (born 28 March 1948) is a Finnish mathematician working in geometric measure theory, complex analysis and harmonic analysis. He is Professor of Mathematics in the Department of Mathematics and Statistics at the University of Helsinki, Finland. He is known for his work on geometric measure theory and in particular applications to complex analysis and harmonic analysis. His work include a counterexample to the general Vitushkin's conjecture and with Mark Melnikov and Joan Verdera he introduced new techniques to understand the geometric structure of removable sets for complex analytic functions which together with other works in the field eventually led to the solution of Painlevé's problem by Xavier Tolsa. His book ''Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability'' is now a widely cited and a standard textbook in this field. Mattila has been the leading figure on creating the geometric measure theory school in Finland a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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