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.^{[1]}^{[2]}

##
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,

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

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^{[1]} if, for all Borel-measurable subsets *A* and *B* of *X* and all 0 ≤ *λ* ≤ 1,

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

The special case *s* = 0 is the inequality

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

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 −∞"

- $s\leq t{\text{ and ))\mu {\text{ is ))t{\text{-convex))\implies \mu {\text{ is ))s{\text{-convex))$

or, equivalently

- $s\leq t\implies \{s{\text{-convex measures))\}\supseteq \{t{\text{-convex measures))\}.$

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 **R**^{n} in the sense above is closely related to the convexity of its probability density function.^{[2]} 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 $e_{s,k}\circ \rho \colon \mathbb {R} ^{k}\to \mathbb {R}$ is a convex function, where

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

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.)^{[1]}