Borel measureIn mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Let be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure defined on the σ-algebra of Borel sets.
Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
Complete measureIn mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure.
Normal distributionIn statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Lebesgue measureIn 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 n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration.
Lebesgue integrationIn mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
Null setIn mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.
Outer measureIn the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
Multivariate normal distributionIn probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.
