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Measure Function



  1. Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6. 


  • (Wikipedia, 2011-Jun-19) ⇒
    • In the mathematical branch measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, volume. A particularly important example is the Lebesgue measure on an Euclidean space, which assigns the conventional length, area and volume of Euclidean geometry to suitable subsets of an n-dimensional Euclidean space Rn, n = 1, 2, 3, .... 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.
    • Let Σ be a σ-algebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
      • Non-negativity: [math]\mu(E)\geq 0[/math] for all [math]E\in\Sigma.[/math]
      • Countable additivity (or σ-additivity): For all countable collections [math]\{E_i\}_{i\in I}[/math] of pairwise disjoint sets in Σ: [math] \mu\Bigl(\bigcup_{i \in I} E_i\Bigr) = \sum_{i \in I} \mu(E_i).[/math]
      • Null empty set: [math]\mu(\varnothing)=0.[/math]
    • Requiring the empty set to have measure zero can be viewed a special case of countable additivity, if one regards the union over an empty collection to be the empty set [math]\bigcup_{\varnothing}=\varnothing[/math] and the sum over an empty collection to be zero [math]\sum_{\varnothing} = 0[/math].


    • 3. Measures and Proportions. Sets of rational numbers, and hence sets of nonnegative integers, are very important examples of scales. We will focus on sets in which 0 is the smallest element. If e is the "lt" relation between x and y and s1 is a set of numbers containing 0 but no smaller number, then there is a nonnegative numeric scale s with s1 as its set and e as its partial ordering. … Suppose we have two points x and y on a scale s1 which has a measure. Then the proportion of x to y is the fraction whose numerator and denominator are the numbers the measure maps x and y into, respectively. … In more conventional notation, if m is a measure function mapping s1 into a nonnegative numeric scale, then the proportion f of x to y is given by "f = m(x)/m(y)". … Thus, we can talk about the proportion of one point on a numeric scale to another, via the identity measure.