mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically in

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

, the trivial measure on any

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...

(''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 if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

''ν''(''X'') = 0. * ''μ'' is an

invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, an ...

(and hence a

quasi-invariant measure In 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 of ''T''. An important class of examples occ ...

) for any

measurable function 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 di ...

''f'' : ''X'' → ''X''. Suppose that ''X'' is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

and that Σ is the Borel ''σ''-algebra on ''X''. * ''μ'' trivially satisfies the condition to be a

regular measure In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topologi ...

. * ''μ'' is never a

strictly positive measure In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points". Definition Let (X, T) be a Hausdorff topological space and let \Sigma be ...

, 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 measure In mathematics, a locally finite measure is a Measure (mathematics), measure for which every point of the measure space has a Neighbourhood (mathematics), neighbourhood of Finite set, finite measure. Definition Let (X, T) be a Hausdorff space, Hau ...

. * If ''X'' is a Hausdorff topological space with its Borel ''σ''-algebra, then ''μ'' trivially satisfies the condition to be a

tight measure In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity". Definitions Let (X, T) be a Hausdorff space, and let \Sigma be a σ-algebra on X that contai ...

. Hence, ''μ'' is also a

Radon measure In mathematics (specifically in measure theory), 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 finite on all compact sets, outer regular on all Borel se ...

. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on ''X''. * If ''X'' is an

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

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 measure In mathematics, there is a folklore claim that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. The theorem this refers to states that there is no translationally invariant measure on a separable Banach space - beca ...

. * If ''X'' is ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

R^''n'' with its usual ''σ''-algebra and ''n''-dimensional

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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

''λ''^''n'', ''μ'' is a

singular measure In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on ...

with respect to ''λ''^''n'': simply decompose R^''n'' as ''A'' = R^''n'' \ and ''B'' = and observe that ''μ''(''A'') = ''λ''^''n''(''B'') = 0.

References

{{Measure theory Measures (measure theory)