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,

$M_{s,\lambda }(u,v)=\{\begin{array}{cc}(\lambda u^s+(1-\lambda )v^s{)}^{1/s}& \text{if }-\mathrm{\infty }<s<0,\\ \min (u,v)& \text{if }s=-\mathrm{\infty },\\ u^{\lambda }{v}^{1-\lambda }& \text{if }s=0.\end{array}$

For subsets A and B of X, we write

$\lambda A+(1-\lambda )B=\{\lambda x+(1-\lambda )y\mid x\in A,y\in B\}$

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,

$\mu (\lambda A+(1-\lambda )B)\ge M_{s,\lambda }(\mu (A),\mu (B)).$

The special case s = 0 is the inequality

$\mu (\lambda A+(1-\lambda )B)\ge \mu (A{)}^{\lambda }\mu (B{)}^{1-\lambda },$

i.e.

$\log \mu (\lambda A+(1-\lambda )B)\ge \lambda \log \mu (A)+(1-\lambda )\log \mu (B).$

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 −∞"

$s\le t\text{ and }\mu \text{ is }t\text{-convex}⟹\mu \text{ is }s\text{-convex}$

or, equivalently

$s\le t⟹\{s\text{-convex measures}\}\supseteq \{t\text{-convex measures}\}.$

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 Rⁿ 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 $e_{s,k}\circ \rho :{ℝ}^k\to ℝ$ is a convex function, where

$e_{s,k}(t)=\{\begin{array}{cc}{t}^{s/(1-sk)}& \text{if }-\mathrm{\infty }<s<0\\ t^{-1/k}& \text{if }s=-\mathrm{\infty },\\ -\log t& \text{if }s=0.\end{array}$

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,

${\mu }_{\ast }(G)=\sup \{\mu (K)\mid K\subseteq G\text{ and }K\text{ is compact}\},$

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.)