“Convex measure”的意思、由来-开放百科全书

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,

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,

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 −∞"

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

is a convex function, where

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,

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]

References

1. ^¹²{{cite journal| last = Borell| first = Christer| title = Convex measures on locally convex spaces| journal = Ark. Mat.| volume = 12| issue = 1–2| year = 1974| pages = 239–252| issn = 0004-2080| doi = 10.1007/BF02384761}}
2. ^¹{{cite journal| last = Borell| first = Christer| title = Convex set functions in d-space| journal = Period. Math. Hungar.| volume = 6| year = 1975| issue = 2| pages = 111–136| issn = 0031-5303| doi = 10.1007/BF02018814}}