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In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure one that is "nowhere zero", or that it is zero "only on points". Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a measure is a function that assigns a number, e. ...
Definition
Let (X, T) be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T (so that every open set is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on (X, Σ) is called strictly positive if every non-empty open subset of X has strictly positive measure. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, a measure is a function that assigns a number, e. ...
In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ...
In more condensed notation, μ is strictly positive if and only if It has been suggested that this article or section be merged with Logical biconditional. ...
Examples - Counting measure on any set X (with any topology) is strictly positive.
- Dirac measure is usually not strictly positive unless the topology T is particularly "coarse" (contains "few" sets). For example, δ0 on the real line R with its usual Borel topology and σ-algebra is not strictly positive; however, if R is equipped with the trivial topology T = {∅, R}, then δ0 is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
- Gaussian measure on Euclidean space Rn (with its Borel topology and σ-algebra) is strictly positive.
- Wiener measure on the space of continuous paths in Rn is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
- Lebesgue measure on Rn (with its Borel topology and σ-algebra) is strictly positive.
- The trivial measure is never strictly positive, regardless of the space or topology used.
In mathematics, the counting measure is an intuitive way to put a measure on any set: the size of a subset is taken to be the number of the subsets elements if this is finite, and ∞ if the subset is infinite. ...
In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that δx({x}) = 1 and in general δx(Y) = 0 for any subset Y of X not containing x, δx(Z) = 1 for any...
In mathematics, the real line is simply the set of real numbers. ...
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space , closely related to the normal distribution in statistics. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ...
In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...
Properties - If μ and ν are two measures on a measurable topological space (X, Σ), with μ strictly positive and also absolutely continuous with respect to ν, then ν is strictly positive as well. The proof is simple: let U ⊆ X be an arbitrary open set; since μ is strictly positive, μ(U) > 0; by absolute continuity, ν(U) > 0 as well.
- Hence, strict positivity is an invariant with respect to equivalence of measures.
Absolute continuity of real functions In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n...
In mathematics, an invariant is something that does not change under a set of transformations. ...
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