EQUIVALENCE (MEASURE THEORY)

In mathematics, and specifically in measure theory, 'equivalence' is a notion of two measures being "the same". Two measures are equivalent if they have the same null sets.

Contents

Definition

Examples

Invariants of measures

References

Definition

Let (''X'', Σ) be a measurable space, and let ''μ'', ''ν'' : Σ → [0, +∞] be two measures. Then ''μ'' is said to be 'equivalent' to ''ν'' if
:$mu\; (A)\; =\; 0\; iff$
u (A) = 0
for measurable sets ''A'' in Σ, i.e. the two measures have precisely the same null sets. Equivalence is often denoted $displaystyle\{mu\; sim$
u} or
u.
In terms of absolute continuity of measures, two measures are equivalent if and only if each is absolutely continuous with respect to the other:
:$mu\; sim$
u iff mu ll
u ll mu.
Equivalence of measures is an equivalence relation on the set of all measures Σ → [0, +∞].

Examples

★ Gaussian measure and Lebesgue measure on the real line are equivalent to one another.

★ Lebesgue measure and Dirac measure on the real line are inequivalent.

Invariants of measures

As is usual in mathematics, one can consider invariants of measures: these are properties of measures defined on a given measurable space such that, if some measure ''μ'' has the property, so do all the other measures to which it is equivalent. More formally, a property ''P'' of measures on (''X'', Σ) is an invariant if
:$left(\; mu\; sim$
u mbox{ and } P(mu)
ight) implies P(
u).
For example, strict positivity is an invariant of measures defined on a topological space (''X'', ''T'') with its Borel σ-algebra.

References