Convex Measure

In measure and probability theory in mathematics, a convex measure is a probability measure that — loosely put — does not assign more mass to any intermediate set "between" two measurable sets ''A'' and ''B'' than it does to ''A'' or ''B'' individually. There are multiple ways in which the comparison between the probabilities of ''A'' and ''B'' and the intermediate set can be made, leading to multiple definitions of convexity, such as log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex spaces in the 1970s.

General definition and special cases

Let ''X'' be a locally convex Hausdorff vector space, and consider a probability measure ''μ'' on the Borel ''σ''-algebra of ''X''. Fix −∞ ≤ ''s'' ≤ 0, and define, for ''u'', ''v'' ≥ 0 and 0 ≤ ''λ'' ≤ 1,
:$M_(u, v) = \begin (\lambda u^s + (1 - \lambda) v^)^ & \text - \infty < s < 0, \\ \min(u, v) & \text s = - \infty, \\ u^ v^ & \text s = 0. \end$
For subsets ''A'' and ''B'' of ''X'', we write
:$\lambda A + (1 - \lambda) B = \$
for their Minkowski sum. With this notation, the measure ''μ'' is said to be ''s''-convex if, for all Borel-measurable subsets ''A'' and ''B'' of ''X'' and all 0 ≤ ''λ'' ≤ 1,
:$\mu(\lambda A + (1 - \lambda) B) \geq M_(\mu(A), \mu(B)).$

The special case ''s'' = 0 is the inequality
:$\mu(\lambda A + (1 - \lambda) B) \geq \mu(A)^ \mu(B)^,$
i.e.
:$\log \mu(\lambda A + (1 - \lambda) B) \geq \lambda \log \mu(A) + (1 - \lambda) \log \mu(B).$
Thus, a measure being 0-convex is the same thing as it being a logarithmically concave measure.

Properties

The classes of ''s''-convex measures form a nested increasing family as ''s'' decreases to −∞"
:$s \leq t \text \mu \text t \text \implies \mu \text s \text$
or, equivalently
:$s \leq t \implies \ \supseteq \.$
Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class.

The convexity of a measure ''μ'' on ''n''-dimensional Euclidean space R^''n'' in the sense above is closely related to the convexity of its probability density function. Indeed, ''μ'' is ''s''-convex if and only if there is an absolutely continuous measure ''ν'' with probability density function ''ρ'' on some R^''k'' so that ''μ'' is the push-forward on ''ν'' under a linear or affine map and $e_ \circ \rho \colon \mathbb^ \to \mathbb$ is a convex function, where
:$e_(t) = \begin t^ & \text -\infty < s < 0 \\ t^ & \text s = - \infty, \\ - \log t & \text s = 0.\end$

Convex measures also satisfy a zero-one law: if ''G'' is a measurable additive subgroup of the vector space ''X'' (i.e. a measurable linear subspace), then the inner measure of ''G'' under ''μ'',
:$\mu_(G) = \sup \,$
must be 0 or 1. (In the case that ''μ'' is a Radon measure, and hence inner regular, the measure ''μ'' and its inner measure coincide, so the ''μ''-measure of ''G'' is then 0 or 1.)

References

{{Measure theory

Measures (measure theory)