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Measure (mathematics) Informally, a measure has the property of being monotone in the sense that if is a subset of the measure of is less than or equal to the measure of Furthermore, the measure of the empty set is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.
Function for which the preimage of a measurable set is measurable. In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition ...
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. [1][2] In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. [3] The scope and application of measurement are ...
Space (mathematics) In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces ...
Measurable space. In mathematics, a measurable space or Borel space[1] is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, and volume with a set of 'points' in the space, but regions of the space are the ...
Lebesgue measure. In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n -spaces. For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume.
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and. A set function generally aims to measure subsets in some way. Measures are typical examples of "measuring" set ...