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convex measure wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 general definition and special cases 2 properties 3 references toggle the table of contents convex measure add languages add links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia 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 edit 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 λ u v λ u s 1 λ v s 1 s if s 0 min u v if s u λ v 1 λ if s 0 displaystyle 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 λ a 1 λ b λ x 1 λ y x a y b displaystyle 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 μ λ a 1 λ b m s λ μ a μ b displaystyle mu lambda a 1 lambda b geq m_ s lambda mu a mu b the special case s 0 is the inequality μ λ a 1 λ b μ a λ μ b 1 λ displaystyle mu lambda a 1 lambda b geq mu a lambda mu b 1 lambda i e log μ λ a 1 λ b λ log μ a 1 λ log μ b displaystyle 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 edit the classes of s convex measures form a nested increasing family as s decreases to s t and μ is t convex μ is s convex displaystyle s leq t text and mu text is t text convex implies mu text is s text convex or equivalently s t s convex measures t convex measures displaystyle 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 ρ r k r displaystyle e_ s k circ rho colon mathbb r k to mathbb r is a convex function where e s k t t s 1 s k if s 0 t 1 k if s log t if s 0 displaystyle 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 μ g sup μ k k g and k is compact displaystyle 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 references edit 1 2 3 borell christer 1974 convex measures on locally convex spaces ark mat 12 1 2 239 252 doi 10 1007 bf02384761 issn 0004 2080 1 2 borell christer 1975 convex set functions in d space period math hungar 6 2 111 136 doi 10 1007 bf02018814 issn 0031 5303 v t e measure theory basic concepts absolute continuity of measures lebesgue integration l p spaces measure measure space probability space measurable space function sets almost everywhere atom baire set borel set equivalence relation borel space carathéodory s criterion cylindrical σ algebra cylinder set 𝜆 system essential range infimum supremum locally measurable π system σ algebra non measurable set vitali set null set support transverse measure universally measurable types of measures atomic baire banach besov borel brown complex complete content logarithmically convex decomposable discrete equivalent finite inner quasi invariant locally finite maximising metric outer outer perfect pre measure sub probability projection valued radon random regular borel regular inner regular outer regular saturated set function σ finite s finite signed singular spectral strictly positive tight vector particular measures counting dirac euler gaussian haar harmonic hausdorff intensity lebesgue infinite dimensional logarithmic product projections pushforward spherical tangent trivial young maps measurable function bochner strongly weakly convergence almost everywhere of measures in measure of random variables in distribution in probability cylinder set measure random compact set element measure process variable vector projection valued measure main results carathéodory s extension theorem convergence theorems dominated monotone vitali decomposition theorems hahn jordan maharam s egorov s fatou s lemma fubini s fubini tonelli hölder s inequality minkowski inequality radon nikodym riesz markov kakutani representation theorem other results disintegration theorem lifting theory lebesgue s density theorem lebesgue differentiation theorem sard s theorem vitali hahn saks theorem for lebesgue measure isoperimetric inequality brunn minkowski theorem milman s reverse minkowski steiner formula prékopa leindler inequality vitale s random brunn minkowski inequality applications related convex analysis descriptive set theory probability theory real analysis spectral theory retrieved from https en wikipedia org w index php title convex_measure oldid 1061678988 category measures measure theory this page was last edited on 23 december 2021 at 05 54 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the 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