In probability theory, a random variable is a measurable function from a probability space to a measurable space of values that the variable can take on.
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If an algebra over a set is closed under countable unions, it is called a sigma algebra and the corresponding field of sets is called a "'measurable space " '.
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In formal notation, we can make any set " X " into a measurable space by taking the sigma-algebra \ Sigma of measurable subsets to consist of all subsets of X.
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It has the advantage of working also for complex-valued random variables or for random variables taking values in any measurable space ( which includes topological spaces endowed by appropriate ?-algebras ).
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Then the counting measure \ mu on this measurable space ( X, \ Sigma ) is the positive measure \ Sigma \ rightarrow [ 0, + \ infty ] defined by
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Every Borel set ( in particular, every closed set and every open set ) in a Euclidean space ( and more generally, in a complete separable metric space ) is a standard measurable space.
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For example, a differentiable manifold ( called also smooth manifold ) is much more geometric than a measurable space, but no one calls it " differentiable space " ( nor " smooth space " ).
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However, the article is abortive, almost pointless, and after all, the last line is meaningless : the subtraction operation is not defined in a measurable space . talk ) 20 : 54, 30 December 2008 ( UTC)
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Moreover, it can be proved that " T " is an ergodic transformation of the measurable space " I " endowed with the probability measure " & mu; " ( this is the hard part of the proof ).
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The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic pushforward of " g " : in essence, the transfer operator is the direct image functor in the category of measurable spaces.
