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.

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 $mu\; approx\; u$.

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