Measure Function

A Measure Function is a well-behaved scalar-output function between measurable spaces.

• AKA: μ, Set Measure Function, Measurable Function.
• Context:
  • range: a Measure Value.
  • It can be associated to a Measurement System.
  • … Scoring Function?
  • It can range from being a Metric Function to being a Non-Metric Measure Function.
  • It can range from being a Static Measure Function to being a Temporal Measure Function.
  • ...
• Example(s):
  • a Distance Metric, such as a set distance measure or a semantic similarity measure.
  • a Random Variable?
  • an Entropy Measure.
  • a Set Measure Function.
  • an Economic Measure, such as a worker productivity measure.
  • a Change Rate Measure, such as a set growth measure.
  • a Relationship Measure, such as a semantic relationship measure.
  • …
• Counter-Example(s):
  • A Non-Measure Function.
• See: Sigma-Algebra,

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References

2013

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Measurable_function
  • In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces.

    This definition can be deceptively simple, however, as special care must be taken regarding the σ-algebras involved. In particular, when a function f: R → R is said to be Lebesgue measurable what is actually meant is that [math]\displaystyle{ f : (\mathbf{R}, \mathcal{L}) \to (\mathbf{R}, \mathcal{B}) }[/math] is a measurable function — that is, the domain and range represent different σ-algebras on the same underlying set (here [math]\displaystyle{ \mathcal{L} }[/math] is the sigma algebra of Lebesgue measurable sets, and [math]\displaystyle{ \mathcal{B} }[/math] is the Borel algebra on R). As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable.

    By convention a topological space is assumed to be equipped with the Borel algebra generated by its open subsets unless otherwise specified. Most commonly this space will be the real or complex numbers. For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously. In practice, some authors use measurable functions to refer only to real-valued measurable functions with respect to the Borel algebra. If the values of the function lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability and Bochner measurability.

    In probability theory, the sigma algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.

2011

• (Wikipedia, 2011-Jun-19) ⇒ http://en.wikipedia.org/wiki/Measure_(mathematics)
  • 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]\displaystyle{ \mu(E)\geq 0 }[/math] for all [math]\displaystyle{ E\in\Sigma. }[/math]
    • Countable additivity (or σ-additivity): For all countable collections [math]\displaystyle{ \{E_i\}_{i\in I} }[/math] of pairwise disjoint sets in Σ: [math]\displaystyle{ \mu\Bigl(\bigcup_{i \in I} E_i\Bigr) = \sum_{i \in I} \mu(E_i). }[/math]
    • Null empty set: [math]\displaystyle{ \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]\displaystyle{ \bigcup_{\varnothing}=\varnothing }[/math] and the sum over an empty collection to be zero [math]\displaystyle{ \sum_{\varnothing} = 0 }[/math].

2007

• http://www.isi.edu/~hobbs/bgt-arithmetic.text
  • 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.

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