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Kuratowski–Ulam Theorem
In mathematics, the Kuratowski–Ulam theorem, introduced by , called also the Fubini theorem for Meagre set#Terminology, category, is an analog of Fubini's theorem for arbitrary second countable Baire spaces. Let ''X'' and ''Y'' be second countable Baire spaces (or, in particular, Polish spaces), and let A \subset X \times Y. Then the following are equivalent if ''A'' has the Property of Baire, Baire property: # ''A'' is Meager set, meager (respectively comeager). # The Set (mathematics), set \ is comeager in X, where A_x=\pi_Y[A\cap \lbrace x \rbrace \times Y], where \pi_Y is the projection onto ''Y''. Even if ''A'' does not have the Baire property, 2. follows from 1. Note that the theorem still holds (perhaps vacuously) for ''X'' an arbitrary Hausdorff space and ''Y'' a Hausdorff space with countable π-base. The theorem is analogous to the regular Fubini's theorem for the case where the considered Function (mathematics), function is a Indicator function, characteristic function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
