Equivalence (measure theory)

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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.

1 Definition
2 Examples
3 Invariants of measures
4 References

[edit] 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

$\mu (A) = 0 \iff \nu (A) = 0$

for measurable sets A in Σ, i.e. the two measures have precisely the same null sets. Equivalence is often denoted $\displaystyle{\mu \sim \nu}$ or $\mu \approx \nu$ .

In terms of absolute continuity of measures, two measures are equivalent if and only if each is absolutely continuous with respect to the other:

$\mu \sim \nu \iff \mu \ll \nu \ll \mu.$

Equivalence of measures is an equivalence relation on the set of all measures Σ → [0, +∞].

[edit] 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.

[edit] 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

$\left( \mu \sim \nu \mbox{ and } P(\mu) \right) \implies P(\nu).$

For example, strict positivity is an invariant of measures defined on a topological space (X, T) with its Borel σ-algebra.

[edit] References

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