In mathematics, specifically in measure theory, 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 ...

(''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 bi ...

''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, ...

(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 o ...

) 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 i ...

''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 po ...

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 topolo ...

. * ''μ'' is never a strictly positive measure, 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 for which every point of the measure space has a neighbourhood of finite measure. Definition Let (X, T) be a Hausdorff topological space and let \Sigma be a \sigma-algebra on X that contains ...

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

. 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) Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...

dimension In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coor ...

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 ve ...

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. * 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

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 ...

''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 ...

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)