Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Measure_(mathematics)> ?p ?o }
- Measure_(mathematics) abstract "In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word, specifically, 1.Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.".
- Measure_(mathematics) thumbnail Measure_illustration.png?width=300.
- Measure_(mathematics) wikiPageExternalLink mt.htm.
- Measure_(mathematics) wikiPageExternalLink UWEETR-2006-0008.pdf.
- Measure_(mathematics) wikiPageID "19873".
- Measure_(mathematics) wikiPageLength "19988".
- Measure_(mathematics) wikiPageOutDegree "163".
- Measure_(mathematics) wikiPageRevisionID "676612909".
- Measure_(mathematics) wikiPageWikiLink Abelian_von_Neumann_algebra.
- Measure_(mathematics) wikiPageWikiLink Almost_everywhere.
- Measure_(mathematics) wikiPageWikiLink Analysis_(mathematics).
- Measure_(mathematics) wikiPageWikiLink Andrey_Kolmogorov.
- Measure_(mathematics) wikiPageWikiLink Angle.
- Measure_(mathematics) wikiPageWikiLink Area.
- Measure_(mathematics) wikiPageWikiLink Axiom_of_choice.
- Measure_(mathematics) wikiPageWikiLink Axiomatic_system.
- Measure_(mathematics) wikiPageWikiLink Axiomatisation.
- Measure_(mathematics) wikiPageWikiLink Baire_measure.
- Measure_(mathematics) wikiPageWikiLink Banach_limit.
- Measure_(mathematics) wikiPageWikiLink Banach_space.
- Measure_(mathematics) wikiPageWikiLink Banach_spaces.
- Measure_(mathematics) wikiPageWikiLink Banach–Tarski_paradox.
- Measure_(mathematics) wikiPageWikiLink Borel_measure.
- Measure_(mathematics) wikiPageWikiLink Canonical_ensemble.
- Measure_(mathematics) wikiPageWikiLink Carathxc3xa9odorys_extension_theorem.
- Measure_(mathematics) wikiPageWikiLink Category:Measure_theory.
- Measure_(mathematics) wikiPageWikiLink Category:Measures_(measure_theory).
- Measure_(mathematics) wikiPageWikiLink Category_(mathematics).
- Measure_(mathematics) wikiPageWikiLink Closed_interval.
- Measure_(mathematics) wikiPageWikiLink Complement_(set_theory).
- Measure_(mathematics) wikiPageWikiLink Complex_measure.
- Measure_(mathematics) wikiPageWikiLink Complex_number.
- Measure_(mathematics) wikiPageWikiLink Conical_combination.
- Measure_(mathematics) wikiPageWikiLink Conservation_law.
- Measure_(mathematics) wikiPageWikiLink Conservation_law_(physics).
- Measure_(mathematics) wikiPageWikiLink Conserved_quantity.
- Measure_(mathematics) wikiPageWikiLink Content_(measure_theory).
- Measure_(mathematics) wikiPageWikiLink Continuous_function.
- Measure_(mathematics) wikiPageWikiLink Countable.
- Measure_(mathematics) wikiPageWikiLink Countable_set.
- Measure_(mathematics) wikiPageWikiLink Countably.
- Measure_(mathematics) wikiPageWikiLink Counting_measure.
- Measure_(mathematics) wikiPageWikiLink Daniell_integral.
- Measure_(mathematics) wikiPageWikiLink Dimension.
- Measure_(mathematics) wikiPageWikiLink Dimension_(mathematics_and_physics).
- Measure_(mathematics) wikiPageWikiLink Dirac_delta_function.
- Measure_(mathematics) wikiPageWikiLink Dirac_measure.
- Measure_(mathematics) wikiPageWikiLink Disjoint_sets.
- Measure_(mathematics) wikiPageWikiLink Dynamical_system.
- Measure_(mathematics) wikiPageWikiLink Empty_set.
- Measure_(mathematics) wikiPageWikiLink Ergodic_measure.
- Measure_(mathematics) wikiPageWikiLink Ergodic_theory.
- Measure_(mathematics) wikiPageWikiLink Ergodicity.
- Measure_(mathematics) wikiPageWikiLink Euclidean_geometry.
- Measure_(mathematics) wikiPageWikiLink Euclidean_space.
- Measure_(mathematics) wikiPageWikiLink Euler_characteristic.
- Measure_(mathematics) wikiPageWikiLink Extended_real_number_line.
- Measure_(mathematics) wikiPageWikiLink Extensive_property.
- Measure_(mathematics) wikiPageWikiLink Fatous_lemma.
- Measure_(mathematics) wikiPageWikiLink Fubinis_theorem.
- Measure_(mathematics) wikiPageWikiLink Function_(mathematics).
- Measure_(mathematics) wikiPageWikiLink Function_composition.
- Measure_(mathematics) wikiPageWikiLink Functional_analysis.
- Measure_(mathematics) wikiPageWikiLink Fuzzy_measure_theory.
- Measure_(mathematics) wikiPageWikiLink Gaussian_measure.
- Measure_(mathematics) wikiPageWikiLink Geometric_measure_theory.
- Measure_(mathematics) wikiPageWikiLink Gibbs_measure.
- Measure_(mathematics) wikiPageWikiLink Gravitational_potential.
- Measure_(mathematics) wikiPageWikiLink Gravity_potential.
- Measure_(mathematics) wikiPageWikiLink Haar_measure.
- Measure_(mathematics) wikiPageWikiLink Hausdorff_measure.
- Measure_(mathematics) wikiPageWikiLink Hausdorff_paradox.
- Measure_(mathematics) wikiPageWikiLink Henri_Lebesgue.
- Measure_(mathematics) wikiPageWikiLink Hilbert_space.
- Measure_(mathematics) wikiPageWikiLink Hyperbolic_angle.
- Measure_(mathematics) wikiPageWikiLink Ideal_(set_theory).
- Measure_(mathematics) wikiPageWikiLink Image_(mathematics).
- Measure_(mathematics) wikiPageWikiLink Indicator_function.
- Measure_(mathematics) wikiPageWikiLink Inner_measure.
- Measure_(mathematics) wikiPageWikiLink Integer.
- Measure_(mathematics) wikiPageWikiLink Integral.
- Measure_(mathematics) wikiPageWikiLink Intensive_and_extensive_properties.
- Measure_(mathematics) wikiPageWikiLink Intersection_(set_theory).
- Measure_(mathematics) wikiPageWikiLink Interval_(mathematics).
- Measure_(mathematics) wikiPageWikiLink Johann_Radon.
- Measure_(mathematics) wikiPageWikiLink Jordan_measure.
- Measure_(mathematics) wikiPageWikiLink Lebesgue_integral.
- Measure_(mathematics) wikiPageWikiLink Lebesgue_integration.
- Measure_(mathematics) wikiPageWikiLink Lebesgue_measurable.
- Measure_(mathematics) wikiPageWikiLink Lebesgue_measure.
- Measure_(mathematics) wikiPageWikiLink Length.
- Measure_(mathematics) wikiPageWikiLink Lifting_theory.
- Measure_(mathematics) wikiPageWikiLink Lindelöf_space.
- Measure_(mathematics) wikiPageWikiLink Linear_combination.
- Measure_(mathematics) wikiPageWikiLink Liouvilles_theorem_(Hamiltonian).
- Measure_(mathematics) wikiPageWikiLink Locally_compact_space.
- Measure_(mathematics) wikiPageWikiLink Locally_convex_space.
- Measure_(mathematics) wikiPageWikiLink Locally_convex_topological_vector_space.
- Measure_(mathematics) wikiPageWikiLink Lorentz_space.
- Measure_(mathematics) wikiPageWikiLink Lp_space.