Suitable for advanced undergraduates and graduate students in mathematics, the treatment covers recurrence, mean and pointwise convergence, ergodic theorem, measure algebras, and automorphisms of compact groups. Additional topics include weak topology and approximation, uniform topology and approximation, invariant measures, unsolved problems, and other subjects.
Suitable for advanced undergraduates and graduate students in mathematics, the treatment covers recurrence, mean and pointwise convergence, ergodic theorem, measure algebras, and automorphisms of compact groups. Additional topics include weak topology and approximation, uniform topology and approximation, invariant measures, unsolved problems, and other subjects.
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Overview
Suitable for advanced undergraduates and graduate students in mathematics, the treatment covers recurrence, mean and pointwise convergence, ergodic theorem, measure algebras, and automorphisms of compact groups. Additional topics include weak topology and approximation, uniform topology and approximation, invariant measures, unsolved problems, and other subjects.
Product Details
| ISBN-13: | 9780486826844 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 11/15/2017 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 112 |
| File size: | 3 MB |
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Read an Excerpt
CHAPTER 1
EXAMPLES
The basic concept is that of a measure space, i.e., a set X together with a specified sigma-algebra of subsets of X and a measure defined on that algebra. Recall that a sigma-algebra is a class of sets closed under the formation of complements and countable unions, and that a measure is a non-negative (possibly infinite) and countably additive set function. The sets in the domain of the measure are called the measurable subsets of X. All the measurs spaces we shall consider will be assumed to be sigma-finite; we shall assume, in other words, that X is the union of countably many sets of finite measure. The purpose of this assumption is to avoid some possible pathology in connection with the Fubini theorem and the Radon-Nikodym theorem; in the presence of sigma-finiteness those theorems are smoothly applicable.
Here are some typical examples of the sort of measure spaces we shall consider. A finite-dimensional Euclidean space, with Borel measura- bility and Lebesgue measure. The unit interval, with the same definitions of measurability and measure. The set of all sequences x = {xn} of O's and l's, where n ranges over the set of all integers; the measurable sets are the elements of the sigma-algebra generated by sets of the form {x : xn=l}, and the measure is determined by the condition that its value on the intersection of k distinct generating sets is always 1/2k. A locally compact topological group with a countable base, with Borel measurability and Haar measure.
A measurable transformation is a mapping from a measure space into a measure space, such that the inverse image of every measurable set is measurable. A measurable transformation T from X into Y will be called invertible if there exists a measurable transformation S from Y into X such that both ST and TS are equal to the identity transformation (on their respective domains). The transformation S is uniquely determined by T; it is called the inverse of T and it is denoted by T-1.
Most of the measurable transformations we shall consider will be measure-preserving, i.e., they will be such that the inverse image of every measurable set has the same measure as the original set. To be honest, the objects of interest are not really measure-preserving transformations, but equivalence classes of such transformations; two transformations are equivalent if they differ only on a set of measure zero. The usual password that permits the consideration of equivalence classes is "identify"; I propose to identify two measure-preserving transformations if and only if they agree almost everywhere. Observe that if a measure-preserving transformation is in vertible, then its inverse is also measure-preserving. Most of the transformations that have been studied in ergodic theory are invertible measure-preserving transformations of a measure space onto itself.
A typical example of a measurable but not measure-preserving transformation on the real line is given by Tx = 2x; it is easy to verify that m(T-1E) =1/2m (E) for every Borel set E (where m, of course, is the measure under consideration, in this case Lebesgue measure). A closely related transformation on the unit interval is defined by Tx = 2x (mod 1). To be completely explicit, I consider the half-open unit interval [0, 1), and I write Tx = 2x when 1/2, [??] x 1/2, and Tx = 2x when 1/2 [??] x 1. If E = [2/8, 5/8), then T-1E is the union of [2/16, 5/16) and [1/2(2/8 + 1), 1/2(5/8 + 1)), so that m(T-1E) = 3/16 + 3/16 = 3/8 = m (E). Similar considerations prove that m(T-1E) = m(E) whenever E is a half-open interval with dyadically rational end-points, and from there it follows easily that T is measure-preserving. Since T is not one-to-one (in fact it is everywhere two-to-one), and since it cannot be made one-toone by any alteration on a set of measure zero, we have here an example of a measure-preserving transformation that is not invertible. An isomorphic representation of the same transformation (in an as yet undefined but pretty obvious sense) is obtained as follows. Let the measure space be the set of all complex numbers of absolute value 1, with Borel measurability, and with measure so normalized that the measure of an arc is 1/2π times its length; define T by Tz = z2.
A simple example of an invertible measure-preserving transformation on the real line is defined by Tx = x+ 1. More generally, in a finite- dimensional Euclidean space, let c be an arbitrary vector and define T by Tx = x + c. Still more generally, in a locally compact group with a left- invariant Haar measure, let c be an arbitrary element of the group and define T by Tx = cx. A useful special case of this last generalization is obtained by considering the circle group ; in this case the geometric realization of T is the rotation through the angle arg c. An isomorphic representation of this special case can be obtained on the unit interval by selecting a number c between 0 and 1 and writing Tx = x + c (mod 1). Explicitly : Tx = x + c when < 1 — c and Tx = x + c — 1 when 1 — c [??] x 1.
Another cluster of examples is suggested by the transformation defined on two-dimensional Euclidean space by T(x, y) = (2x, 1/2 y). The inverse image of the unit square is a rectangle with base ~ and altitude 2. Since, similarly, the inverse image of every rectangle is a rectangle of the same area, it follows that T is a measure-preserving transformation; obviously T is invertible. In an attempt to generalize this example, consider an arbitrary linear transformation T on a finite-dimensional Euclidean space. The range of T is a subspace whose inverse image is the entire space. Since a proper subspace has measure zero, it follows that in order that T be measure-preserving it is necessary that T be non-singular. If T is non-singular with determinant d, then it is well-known that m(T-1E) = m(E)/|d| for every Borel set E. (This well-known fact is seldom proved. A proof can be given by analytic techniques involving Jacobians; for a direct proof see Carathéodory, Vorlesungen ueber reelle Funktionen, 1927, p. 346.)
The non-singular linear transformations of a real, finite-dimensional vector space can be characterized as the continuous automorphisms of the additive vector group. This suggests the consideration of an arbitrary locally compact group with a left Haar measure and of a continuous automorphism T of that group. To find out whether or not T is measure-preserving, we must compare m(E) with n(E) = m(T-1E). The set function n is obviously a measure ; it is natural to ask if it is a left Haar measure. This means: is m(T-1(xE)) the same as m(T-1E)? The answer isyes, because, in fact, T-1(xE) is a left translate of T-1E; indeed, an obvious computation shows that T-1(xE) = (T-x-1)-1T-1E. It follows from the uniqueness of Haar measure that m(T-1E) is a constant multiple of m(E). In general this is all that we can conclude; the examples of non-singular linear transformations show that an automorphism need not be measure-preserving. If, however, the group X is compact, then m (X) is finite and therefore the constant of proportionality can be evaluated by putting E equal to X; since T-1X=X, it follows that the constant is equal to 1, and hence that T is measure-preserving.
An interesting special case is obtained by considering the group to be the torus, i.e., the Cartesian product of two circles. Concretely, the elements of the group are pairs (u, v) of complex numbers of modulus one; the group operation is coordinatewise multiplication. It is easy to show that the most general continuous automorphism is given by a two-rowed unimodular matrix, i.e., by a matrix with integer entries and determinant [+ or -] 1; if [MATHEMATICAL EXPRESSION OMITTED] is such a matrix, the corresponding automorphism T is defined by T(u, v) = {uavb, ucvd).
Let X be the space of sequences x = {xn}, n = 0, ±z1, ±2, —, described before ; let T be the transformation induced by a unit shift on the indices, i.e., Tx=y = y = {yn}, where yn = xn+1. This transformation is measure-preserving and invertible. If X is the unilateral sequence space, i.e., the elements of X are sequences {xn} with n = 0, 1, 2, ..., the same equation defines a measure-preserving but non-invertible (two-to-one) transformation.
There is a simple mapping S from the unilateral sequence space to the unit interval; S sends the sequence {xn} of 0's and 1's onto the number whose binary expansion is The transformation S is measure-preserving and essentially one-to-one. It is not quite one-to-one, because a dyadically rational number has two possible expansions. The set of sequences whose image is dyadically rational has the same cardinal number as the set of dyadically rational numbers; both sets are countably infinite. If we suitably redefine S on these exceptional sequences, the result is an invertible measure-preserving transformation from the sequence space onto the unit interval. The existence of such a transformation shows that the measure-theoretic structures of the two spaces are isomorphic. The isomorphism (i.e., the transformation S) carries the unilateral shift T onto an invertible measure-preserving transformation T' on the interval; T' is defined by TESTS'1. An examination of the definitions of S and T shows that T is an old friend: T'x = 2x (mod 1) almost everywhere.
There is a natural correspondence between the bilateral sequence space and the Cartesian product of the unilateral sequence space with itself; the correspondence sends {..., x-2, x-1, x0}, x1. x-2} ... onto
[MATHEMATICAL EXPRESSION OMITTED]
This correspondence is easily seen to be an invertible measure-preserving transformation, and, therefore, a measure-theoretic isomorphism. If we denote this isomorphism by P and if we denote by Q the Cartesian product of with itself (so that Q (x, y) = (Sx, Sy) whenever x and y are unilateral sequences), then the composite transformation QP is an isomorphism from the bilateral sequence space onto the unit square. This isomorphism carries the bilateral shift onto an invertible measure-preserving transformation T" on the square. An examination of the definitions shows that T" is a close relative of an old friend; it is given by T" (x, y) = (2x, 1/2y when 0 [??] x 1/2 and by T"(x, y) = (2x, 1/2 (y + 1)) when 1/2 [??] x1. (These equations, valid almost everywhere, must of course be taken modulo 1.) The transformation T" can be described geometrically, as follows. Transform the unit square by the linear transformation that sends (x, y) onto (2x, 1/2 y), getting a rectangle whose bottom edge is [0, 2) and whose left edge is [0, 1/2); cut off the right half of this rectangle (with bottom edge [1, 2)) and move it, by translation, to the top half of the unit square. Since these actions are faintly reminiscent of what happens in kneading dough, the transformation T" is sometimes called the baker's transformation.
CHAPTER 2
RECURRENCE
In order to discuss the asymptotic properties of a measure-preserving transformation T, i.e., the properties of the sequence {Tn}, the powers of T must make sense; for this reason we shall, throughout the sequel, restrict attention to transformations from a set X into itself. The earliest and simplest asymptotic questions were raised by Poincaré (Calcul des probabilités, 1912); they concern recurrence. If T is a measurable transformation on X and E is a measurable subset of X, a point x of E is called recurrent (with respect to E and T) if Tnx [member of] E for at least one positive integer n. Our first result is typical of the subject.
Recurrence Theorem. If t is a measure preserving {but not necessarily invertible) transformation on a space of finite measure, and if E is a measurable set y then almost every point of E is recurrent.
Proof. If not, then the set F of those points of E that never return to E is a set of positive measure. The set F is measurable since
[MATHEMATICAL EXPRESSION OMITTED]
If x [member of] F, then none of the points Tx, T2x, T*x, ..., belongs to F, or, in other words, F is disjoint from T~nF for all positive n. It follows that the sets F, T]IT-1F, T-2F, ..., are pairwise disjoint, since
[MATHEMATICAL EXPRESSION OMITTED]
Since T is measure-preserving and the space has finite measure, this is a contradiction.
The recurrence theorem implies a stronger version of itself. Not only is it true that for almost every x in E at least one term of the sequence Tx, T2x, ... belongs to E; in fact, for almost every x in Ey there are infinitely many values of n such that Tnx [member of] E. The idea of the proof is to apply the recurrence theorem to each power of T. Precisely speaking, if Fn is the set of those points of E that never return to E under the action of Tn, then, by the recurrence theorem, m(Fn) = 0. If x [member of] E-(F1 [union] F2 [union] ...), then Tnx E for some positive n, since x[member of]E-F1(=E — Similarly, since E — Fn it follows that Tkn[member of]E for some positive k. The strengthened version of the recurrence theorem follows by an inductive repetition of this twice-repeated argument.
In the proof of the original recurrence theorem, the measure-preserving character of the transformation and the finiteness of the measure were used in a very weak sense only. All that was essential was the nonexistence of a set F of positive measure such that the sets F, T-1F, T-2F. ... are pairwise disjoint. Motivated by this remark, we introduce a new concept: a measurable transformation T is called dissipative if there exists a measurable set F of positive measure such that the sets F, T-1F, T-2F, ... are pairwise disjoint; in the contrary case T is called conservative. It is clear that the weak recurrence theorem is valid for every conservative transformation.
The proof of the strong recurrence theorem depends on the applicability of the weak theorem to every power of T. It is clear that if T is dissipative, then so is every power of T. This is not good enough. If we knew that every power of a conservative transformation is itself conservative, we could assert the conclusion of the strong recurrence theorem for every conservative transformation.
(Continues…)
Excerpted from "Lectures on Ergodic Theory"
by .
Copyright © 2017 Paul R. Halmos.
Excerpted by permission of Dover Publications, Inc..
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