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In mathematics, a measure is a function that assigns a number, e.g., a "size", "volume", or "probability", to subsets of a given set. The concept is important in mathematical analysis and probability theory.
Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and integrals. It is of importance in probability and statistics.
See also Lebesgue integration, Lebesgue measure
Formally, a countably additive measure μ is a function defined on a sigma algebra of subsets of X with values in the extended interval [0, ∞] such that the following properties are satisfied:
The members of are called measurable sets and the structureA structure can be a building or other thing built, such as a bridge, but here the structure of a thing is how the parts of it relate to each other, how it is put together; how it works is process, but process requires a viable structure. Both reality and is called a measure space. The following properties can be derived from the definition above:
A measure space Ω is called finite if μ(Ω) is a finite real number (rather than ∞). It is called σ-finite if Ω is the countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable unionAbstract algebra Algebra In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. This article uses mathematical symbols. Basic definition If of sets with finite measure.
For example, the real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which sts k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very nice properties; σ-finiteness can be compared in this respect to separability of topological spaces.
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