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| In [[mathematics]] — specifically, in [[ergodic theory]] — a '''maximising measure''' is a particular kind of [[probability measure]]. Informally, a probability measure ''μ'' is a maximising measure for some function ''f'' if the [[Lebesgue integration|integral]] of ''f'' with respect to ''μ'' is “as big as it can be”. The theory of maximising measures is relatively young and quite little is known about their general structure and properties.
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| ==Definition==
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| Let ''X'' be a [[topological space]] and let ''T'' : ''X'' → ''X'' be a [[continuous function]]. Let Inv(''T'') denote the set of all [[Borel measure|Borel]] probability measures on ''X'' that are [[invariant measure|invariant]] under ''T'', i.e., for every Borel-measurable subset ''A'' of ''X'', ''μ''(''T''<sup>−1</sup>(''A'')) = ''μ''(''A''). (Note that, by the [[Krylov-Bogolyubov theorem]], Inv(''T'') is non-empty.) Define, for continuous functions ''f'' : ''X'' → '''R''', the maximum integral function ''β'' by
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| :<math>\beta(f) := \sup \left. \left\{ \int_{X} f \, \mathrm{d} \nu \right| \nu \in \mathrm{Inv}(T) \right\}.</math>
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| A probability measure ''μ'' in Inv(''T'') is said to be a '''maximising measure''' for ''f'' if
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| :<math>\int_{X} f \, \mathrm{d} \mu = \beta(f).</math>
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| ==Properties==
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| * It can be shown that if ''X'' is a [[compact space]], then Inv(''T'') is also compact with respect to the topology of [[weak convergence of measures]]. Hence, in this case, each continuous function ''f'' : ''X'' → '''R''' has at least one maximising measure.
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| * If ''T'' is a continuous map of a compact [[metric space]] ''X'' into itself and ''E'' is a [[topological vector space]] that is [[dense set|densely]] and [[continuously embedded]] in ''C''(''X''; '''R'''), then the set of all ''f'' in ''E'' that have a unique maximising measure is equal to a [[countable set|countable]] [[intersection (set theory)|intersection]] of [[open set|open]] dense subsets of ''E''.
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| ==References==
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| * {{cite book
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| | last = Morris
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| | first = Ian
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| | title = Topics in Thermodynamic Formalism: Random Equilibrium States and Ergodic Optimisation
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| | url = http://www.warwick.ac.uk/staff/Ian.Morris/thesis.ps
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| | format = PostScript
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| | year = 2006
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| | accessdate = 2008-07-05
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| | location = University of Manchester, UK
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| | publisher = Ph.D. thesis
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| }}
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| * {{cite journal
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| | doi = 10.3934/dcds.2006.15.197
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| | last = Jenkinson
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| | first = Oliver
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| | title = Ergodic optimization
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| | journal = Discrete and Continuous Dynamical Systems
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| | volume = 15
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| | year = 2006
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| | issue = 1
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| | pages = 197–224
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| | issn = 1078-0947
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| }} {{MathSciNet|id=2191393}}
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| [[Category:Ergodic theory]]
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| [[Category:Measures (measure theory)]]
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I'm Marianne (18) from Kobenhavn V, Denmark.
I'm learning Arabic literature at a local university and I'm just about to graduate.
I have a part time job in a post office.
Feel free to surf to my blog post - boom beach hack tool no survey no download (click here to find out more)