Limits of inverse systems of measures

Mallory, J. D.; Sion, Maurice

Annales de l'Institut Fourier, Tome 21 (1971) no. 1, pp. 25-57.

Étant donné un système projectif d’espaces mesurés $(X_i,{\mu }_i{)}_{i\in I}$, on étudie le problème d’existence d’une limite projective en considérant d’abord une mesure $\tilde{\mu }$ définie sur le produit ${\prod }_{i\in I}{X}_i$. Sous de simples conditions de régularité des ${\mu }_i$, on montre que $\tilde{\mu }$ a presque toutes les propriétés d’une limite. En outre, la limite projective ${\mu }^{\prime }$ peut exister seulement si $\tilde{\mu }$ est elle-même une “limite” dans un sens plus général et ${\mu }^{\prime }$ est alors la restriction de $\tilde{\mu }$ à l’ensemble limite des $X_i$. On obtient des résultats plus forts que ceux connus jusqu’à présent en examinant cette restriction.

In this paper the problem of the existence of an inverse (or projective) limit measure ${\mu }^{\prime }$ of an inverse system of measure spaces $(X_i,{\mu }_i)$ is approached by obtaining first a measure $\tilde{\mu }$ on the whole product space ${\prod }_{i\in I}{X}_i$.

The measure $\tilde{\mu }$ will have many of the properties of a limit measure provided only that the measures ${\mu }_i$ possess mild regularity properties.

It is shown that ${\mu }^{\prime }$ can only exist when $\tilde{\mu }$ is itself a “limit” measure in a more general sense, and that ${\mu }^{\prime }$ must then be the restriction of $\tilde{\mu }$ to the projective limit set $L$.

Results stronger than those previously known are obtained by examining $\tilde{\mu }$ restricted to $L$.

[1] E. Sparre Andersen and B. Jessen, Of the introduction of measures in infinite product sets, Danski Vid. Selskab Mat. Fys. Medd. 25 (1948) No. 4. | MR | Zbl

[2] S.K. Berberian, Measure and integration, MacMillan Co. New York (1965). | MR | Zbl

[3] S. Bochner, Harmonic analysis and probability theory Univ. of Cal. Press, Berkeley, 1955. | Zbl

[4] N. Bourbaki, Théorie des Ensembles Livre I Ch. III, Hermann, Paris.

[5] J.R. Choksi, inverse limits of measure spaces, Proc. London Math. Soc., 8 (1958) 321-342. | MR | Zbl

[6] P.R. Halmos, Measure theory, Van Nostrand, New York, (1950). | MR | Zbl

[7] Mr. Jirina, Conditional probabilities on strictly separable σ-algebra (Russian) Czech. Math. J. 4 (79) (1954) 372-80. | MR | Zbl

[8] J.F.C. Kingmann and S.J. Taylor, Introduction to measure and probability, Cambridge U.P. London, 1966. | Zbl

[9] A.N. Kolmogoroff, Grundebegriff der Wahrscheinlichheit (Berlin, 1933), (English translation: Chelsea, New York 1956).

[10] D.J. Mallory, Limits of Inverse Systems of Measures, thesis, University of British Columbia 1968.

[11] E. Marczewski, On Compact Measures, Fund. Math. 40 (1953) 113-24. | MR | Zbl

[12] M. Metivier, Limites projectives de mesures, Martingales Applications, Ann. di Mathematica 63, (1963) 225-352. | MR | Zbl

[13] P.A. Meyer, Probabilities and Potentials, Blaisdell (1966). | Zbl

[14] J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, 1965. | MR | Zbl

[15] Yu, V. Prokhorov, Convergence of random processes and limit theorems of probability theory (in Russian), Teoriya Veroyatnostei i ee Primeneniya 1, 177-237 (1956). English translation: Theory Probab. and Appl. 1, 157-214 (1956). | MR | Zbl

[16] J.P. Raoult, Sur une généralisation d'un théorème d'Ionescu Tulcea, C.R. Acad. Sc. Paris 259 (1964), 2769-2772. | MR | Zbl

[17] C.L. Scheffer, Generalizations of the theory of Lebesgue spaces and of the definition of entropy in ergodic theory, thesis, University of Utrecht 1966.

[18] C.L. Scheffer, Projective limits of directed projective systems of probability spaces, Z. Wahrscheinlichkeitstheorie verw. Geb. 13, 60-80 (1969). | MR | Zbl

[19] M. Sion, Lecture Notes on Measure Theory, Biennial Seminar of the Canadian Mathematical Congress, (1965).

[20] M. Sion, Introduction to the Methods of Real Analysis, Holt, Rinehart and Winston, New York (1968). | MR | Zbl