Invariant measure

An invariant measure on a measurable space $ ( X , \mathfrak B ) $ with respect to a measurable transformation $ T $ of this space is a measure $ \mu $ on $ \mathfrak B $ for which $ \mu ( A) = \mu ( T ^ {-} 1 A ) $ for all $ A \in \mathfrak B $. It is usually assumed that the measure is finite (that is, $ \mu ( X) < \infty $) or at least $ \sigma $- finite (that is, $ X $ can be expressed as a countable union $ \cup X _ {n} $, where $ \mu ( X _ {n} ) < \infty $). In the most important case when $ T $ is a bijection and the mapping $ T ^ {-} 1 $ is also measurable (one then says that $ T $ is invertible, having in mind invertibility in the class of measurable transformations), the invariance of the measure $ \mu $ is equivalent to the property that $ \mu ( A) = \mu ( TA ) $ for all $ A \in \mathfrak B $. Finally, an invariant measure for a family of (measurable) transformations, such as a semi-group, a group, a flow, etc., is a measure that is invariant under all the transformations of this family. The notion of an invariant measure plays an important role in the theory of dynamical systems and ergodic theory. In the latter one considers various properties of dynamical systems in a measure space $ ( X , \mathfrak B , \mu ) $ having $ \mu $ as their invariant measure. If a dynamical system has several invariant measures, for example, $ \mu $ and $ \nu $, then its properties as a system in $ ( X , \mathfrak B , \mu ) $( properties with respect to the invariant measure $ \mu $) can differ from its properties as a system in $ ( X , \mathfrak B , \nu ) $( properties with respect to $ \nu $). When one considers different invariant measures for a fixed dynamical system, one often refers to the properties of the system with respect to the invariant measure $ \mu $ as properties of the measure $ \mu $( for example, "m is ergodic measureergodic" means ergodicity of the given system as a system in $ ( X , \mathfrak B , \mu ) $, that is, the absence of invariant sets $ A \in \mathfrak B $ with $ \mu ( A ) > 0 $ and $ \mu ( X \setminus A ) > 0 $).

Historically, the first examples of invariant measures were related to differentiability properties of transformations generating flows of certain special types on smooth manifolds (see Hamiltonian system; Integral invariant). In terms of (local) coordinates $ x _ {1} \dots x _ {n} $ these measures $ \mu $ can be represented in the form $ d \mu = \rho d x _ {1} \dots d x _ {n} $, and there are explicit expressions for the density $ \rho = \rho ( x _ {1} \dots x _ {n} ) $. In examples of algebraic origin (groups of shifts, etc.) the invariant measure is often a Haar measure or a measure obtained from it by some natural construction.

In topological dynamics, N.N. Bogolyubov and I.M. Krylov proved (, see also , ) the existence of finite ergodic invariant measures for continuous flows and cascades on a metric compactum $ X $( various generalizations are possible , , ). Non-ergodic finite invariant measures are in a certain sense linear combinations of ergodic ones; the supports of finite invariant measures are related in a certain way to the behaviour of the trajectories in $ X $( all these invariant measures are concentrated on the so-called minimal centre of attraction ). It is not worth while to look for more detailed statements on the properties of invariant measures in the general case; they can be quite varied. Thus, in one case an ergodic invariant measure can be concentrated at a single point, in another, it can be positive for all open subsets of $ X $ and possess properties of a "quasi-random" nature (mixing, positive entropy, etc.), the description and study of which relate to ergodic theory (whereas reverting to the latter in the previous case would make no sense). There are therefore a number of studies on the existence of invariant measures with various interesting properties for dynamical systems, depending on the type of the latter.

Finally there is a purely metric version of the problem of the existence of invariant measures. Suppose that a dynamical system has a quasi-invariant measure $ \mu $; does it then have an invariant measure $ \nu $ equivalent to $ \mu $? (A discussion of this statement of the question can be found in . Another can be found in .) The answer is negative, in general, even if $ \nu $ is required merely to be $ \sigma $- finite and $ ( X , \mathfrak B , \mu ) $ is a Lebesgue space . Different versions are known of necessary and sufficient conditions for the existence of finite invariant measures; the most successful are the conditions of A. Hajian and S. Kakutani , .

D.V. Anosov

An invariant measure in probability theory is defined with respect to a transition probability (cf. Transition probabilities). Let $ ( X , {\mathcal A} ) $ be a measurable space, where $ {\mathcal A} $ is a $ \sigma $- algebra, and let $ P ( x , A ) $, $ x \in X $, $ A \in {\mathcal A} $, be a transition probability (that is, $ P ( x , \cdot ) $ is a probability measure on $ {\mathcal A} $ for each $ x \in X $ and $ P ( \cdot , A ) $ is $ {\mathcal A} $- measurable for each $ A \in {\mathcal A} $). Then a countably-additive measure $ \mu $ on $ ( X , {\mathcal A} ) $ is said to be invariant with respect to $ P $ if

$$ \mu ( A ) = \int\limits _ { X } P ( x , A ) \mu ( dx ) \ \ \textrm{ for all } A \in {\mathcal A} . $$

If $ T $ is a measurable mapping from $ ( X , {\mathcal A} ) $ into itself, then the measure $ \mu $ is invariant with respect to $ T $ if and only if it is invariant with respect to the transition probability $ P ( x , A ) = \chi _ {T ( x ) } ( A ) $, where $ \chi _ {y} ( A ) = 1 $ for $ y \in A $ and $ \chi _ {y} ( A ) = 0 $ for $ y \notin A $.

V.V. Sazonov

References for both sections follow.

References

[1]	N.N. [N.N. Bogolyubov] Bogoluboff, N.M. [N.M. Krylov] Kriloff, "La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique nonlinéaire" Ann. of Math. , 38 (1937) pp. 65–113
[2]	J. Oxtoby, "Ergodic sets" Bull. Amer. Math. Soc. , 58 (1952) pp. 116–136
[3]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[4]	N.N. Bogolyubov, "On some ergodic properties of continuous transformation groups" , Selected Works , 1 , Kiev (1969) pp. 561–569 (In Russian)
[5]	S.V. Fomin, "Finite measures invariant under flows" Transl. Amer. Math. Soc. (2) , 57 (1966) pp. 113–122 Mat. Sb. , 12 : 1 (1943) pp. 99–108
[6]	S.V. Fomin, "On measures invariant under a group of transformations" Transl. Amer. Math. Soc. (2) , 51 (1966) pp. 317–332 Izv. Akad. Nauk. SSSR Ser. Mat. , 14 : 3 (1950) pp. 261–274
[7]	P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956)
[8]	D.A. Vladimirov, "Boolesche Algebren" , Akademie Verlag (1978) (Translated from Russian)
[9]	D.S. Ornstein, "On invariant measures" Bull. Amer. Math. Soc. , 66 : 4 (1960) pp. 297–300
[10]	A. Hajian, S. Kakutani, "Weakly wandering sets and invariant measures" Trans. Amer. Math. Soc. , 110 (1964) pp. 136–151
[11]	J. Neveu, "Mathematical foundations of the calculus of probability" , Holden-Day (1965) (Translated from French)
[12]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[13]	K. Yosida, "Functional analysis" , Springer (1980)

Comments

For the existence of invariant measures for groups of transformations one may also consult [a1]. As to the "ergodic decompositionergodic decomposition" of an invariant measure (i.e. to obtain it as a linear combination of ergodic invariant measures), this is a straightforward consequence of Choquet theory (cf. Choquet simplex), provided the underlying space is a metric compactum. For more general compact spaces, see [a2].

Apart from [11], [12] and [13], much information on measures invariant with respect to a transition probability is included in [a3] (its appendix contains a result on the possibility of an ergodic decomposition of such measures), [a4] and [a5].

Invariant measures with respect to transition probabilities take a prominent part in the theory of Markov processes (cf. Markov process), mostly in the study of recurrence in the theory of Markov chains (cf. Markov chain).

References

[a1]	S. Glasner, "Proximal flows" , Springer (1976)
[a2]	H.B. Keynes, D. Newton, "The structure of ergodic measures for compact group extensions" Israel J. Math. , 18 (1974) pp. 363–399
[a3]	Y. Kifer, "Ergodic theory of random transformations" , Birkhäuser (1986)
[a4]	U. Krengel, "Ergodic theorems" , de Gruyter (1985) pp. 261
[a5]	D. Revuz, "Markov chains" , North-Holland (1975)