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Lebesgue's Decomposition Theorem
In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based on their relationship with another measure. Definition The theorem states that if (\Omega,\Sigma) is a measurable space and \mu and \nu are σ-finite signed measures on \Sigma, then there exist two uniquely determined σ-finite signed measures \nu_0 and \nu_1 such that: * \nu=\nu_0+\nu_1\, * \nu_0\ll\mu (that is, \nu_0 is absolutely continuous with respect to \mu) * \nu_1\perp\mu (that is, \nu_1 and \mu are singular). Refinement Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue-Radon-Nikodym theorem. That is, let (\Omega,\Sigma) be a measure space, \mu a σ-finite positive measure on \Sigma and \lambda a complex measure on \Sigma. * There is a unique pair of complex measures on \Sigma such that \lambda = \lambda_a + \lambda_s, \quad \lambda_a \ll \mu, \quad \lambda_s \perp \mu. If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
