FACTOID # 21: 15% of Army recruits from South Dakota are Native American, which is roughly the same percentage for female Army recruits in the state.

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Encyclopedia > Borel measure

In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure ba (where a < b). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the Borel algebra (or Borel Ïƒ-algebra) on a topological space X is a Ïƒ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this Ïƒ-algebra: The minimal Ïƒ-algebra containing the open sets. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, a measure is a function that assigns a number, e. ...

The Borel measure is not complete, which is why in practice the complete Lebesgue measure is preferred: every Borel measurable set is also Lebesgue measurable, and the measures of the set agree. In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0). ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

In a more general (abstract) measure-theoretic context, Let E be a Hausdorff space. A measure μ on the σ-algebra $mathfrak{B}(E)$ (the Borel σ-algebra on E) is Borel iff $mu(K) < +infty$ for all $K subset E$ compact. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, the Borel algebra (or Borel Ïƒ-algebra) on a topological space X is a Ïƒ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this Ïƒ-algebra: The minimal Ïƒ-algebra containing the open sets. ...

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