Convex measure explained

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,} (u, v) = \begin{cases} ( λ u ^{s} + ( 1 - λ ) v ^{s} ) ^{1/s} & if - inf ty < s < 0 , \ min (u, v) & if s = - inf ty, \ u ^{λ} v ^{1-} & if s = 0 . \end{cases} For subsets A and B of X, we writeλ A + ( 1 - λ ) B = \{ λ x + ( 1 - λ ) 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 ≤ &lambda; ≤ 1,\mu( λ A + ( 1 - λ ) B) \geq M _{s,} (\mu(A), \mu(B)).

The special case s = 0 is the inequality

\mu( λ A + ( 1 - λ ) B) \geq \mu(A) ^{λ} \mu(B) ^{1} , i.e.log \mu( λ A + ( 1 - λ ) B) \geq λ log \mu(A) + ( 1 - λ ) 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 and \mu is t -convex \implies \mu is s -convex or, equivalentlys \leq t \implies \{ s -convexmeasures \} \supseteq \{ t -convexmeasures \}. 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.^[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,} \circ \rho \colon R ^{k} \to R is a convex function, wheree _{s,} (t) = \begin{cases} t ^{s} & if -inf ty < s < 0 \ t ^{-1/k} & if s = - inf ty, \ - log t & 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 and K iscompact \}, 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]

Notes and References

1. Borell. Christer. Convex measures on locally convex spaces. Ark. Mat.. 12. 1–2. 1974. 239 - 252. 0004-2080. 10.1007/BF02384761. free.
2. Borell. Christer. Convex set functions in d-space. Period. Math. Hungar.. 6. 1975. 2. 111 - 136. 0031-5303. 10.1007/BF02018814.