Trivial measure
Encyclopedia
In mathematics
, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
Suppose that X is a topological space
and that Σ is the Borel σ-algebra on X.
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...
, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
Properties of the trivial measure
Let μ denote the trivial measure on some measurable space (X, Σ).- A measure ν is the trivial measure μ if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
ν(X) = 0. - μ is an invariant measureInvariant measureIn mathematics, an invariant measure is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems...
(and hence a quasi-invariant measureQuasi-invariant measureIn mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function by T...
) for any measurable functionMeasurable functionIn 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...
f : X → X.
Suppose that X is a 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 that Σ is the Borel σ-algebra on X.
- μ trivially satisfies the condition to be a regular measureRegular measureIn mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed".-Definition:...
. - μ is never a strictly positive measureStrictly positive measureIn mathematics, 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:...
, regardless of (X, Σ), since every measurable set has zero measure. - Since μ(X) = 0, μ is always a finite measure, and hence a locally finite measureLocally finite measureIn mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.-Definition:...
. - If X is a HausdorffHausdorff spaceIn 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 with its Borel σ-algebra, then μ trivially satisfies the condition to be an tight measure. Hence, μ is also a Radon measureRadon measureIn mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on X. - If X is an infiniteInfinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
-dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
with its Borel σ-algebra, then μ is the only measure on (X, Σ) that is locally finite and invariant under all translations of X. See the article There is no infinite-dimensional Lebesgue measureThere is no infinite-dimensional Lebesgue measureIn mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite-dimensional spaces: often, the abstract Wiener space construction is used...
. - If X is n-dimensional Euclidean spaceEuclidean spaceIn 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 usual σ-algebra and n-dimensional Lebesgue measureLebesgue measureIn 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...
λn, μ is a singular measureSingular measureIn mathematics, two positive measures μ and ν defined on a measurable space are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A...
with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.