In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that it is zero "only on points".

Definition

Let (X, T) be a Hausdorff

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

 topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 and let Σ be a σ-algebra on X that contains the topology T (so that every open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 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 more condensed notation, μ is strictly positive if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

Examples

• Counting measure
  Counting measure
  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 elements in the subset, if the subset is finite, and ∞ if the subset is infinite....
  
   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
  Real line
  In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
  
   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
  Gaussian measure
  In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...
  
   on Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   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
  Lebesgue measure
  In measure theory, 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...
  
   on Rⁿ (with its Borel topology and σ-algebra) is strictly positive.
• The trivial measure
  Trivial measure
  In mathematics, specifically in measure theory, the trivial measure on any measurable space is the measure μ which assigns zero measure to every measurable set: μ = 0 for all A in Σ.-Properties of the trivial measure:...
  
   is never strictly positive, regardless of the space X or the topology used, except when X is empty.

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
  Invariant (mathematics)
  In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
  
   with respect to equivalence of measures.

See also

• Support (measure theory)
  Support (measure theory)
  In mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...
  
  : a measure is strictly positive if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
  its support is the whole space.