In a note which we received from W. Schachermayer he called attention to some confusion in his paper [S1], which he said was noticed by H. Buhler. Our reading of this note made us unhappily aware that some related confusion occurred in our papers [Sm] and [FS]. The purpose of the present note is to eliminate the latter confusion. We are grateful to Schachermayer for bringing the confusion to our attention. His original note, expanded, will be published as [S2].

This is partly exposition, partly generalization and partly completion of some of Vershik's work towards classifying sequences of measurable partitions.

Let B(t), 0≤t≤∞ be a Brownian motion on (ω, F, P) with with B 0=0. Let F(t), 0≤t≤∞ be its filtration, with F(∞)=F. We construct, simple examples of probability measures P′≈P for which this filtration is not generated by the corresponding Girsanov process, but is nevertheless generated by some process which is a Brownian motion for the measure P′.

We investigate the size of the set of reals which can be represented in base gamma using only the digits 0,1,3. It is shown that this set has Lebesgue measure zero for gamma less than or equal to 1/3 and equals an interval for gamma greater than or equal to 2/5. Our main goal is to prove that it has Lebesgue measure zero for a certain countable subset of (1/3,2/5).

In 1960 Schmidt [S] showed that if p and q are not powers of the same integer, i.e., if log q/log p is irrational, then for certain special measures μ on [0,1), invariant under S:x ↦ px (mod 1), μ-almost every x is normal to the base q. The measures considered in [S] were similar to Cantor-Lebesgue measure: namely, under μ the p-digit process was a special i.i.d. process where for some k ≥ 2 the elements of a certain k-element subset of the p-digits assumed probability 1/k each. The proof was fairly complicated, and did not seem to yield much more (see Keane and Pearce [K] for another proof).

A construction of normal sequences, similar to Champernowne’s one, is obtained for Markov shifts and intrinsically ergodic subshifts. For eachn a set Ω n ofn-blocks is selected. A normal sequence is constructed by first concatenating the blocks of Ω n (in any order) and then concatenating the resultant finite sequences successively.

A general method of constructing block codes between Bernoulli shifts is introduced. This method generalizes an example of Boyle and Tuncel.

It is shown that if the continued fractions of the rationals 12, 13, 23, 14, 24, 34, 15, 25, 35, 45,… are concatenated, a normal continued fraction is obtained.

It is shown that every invertible ergodic transformation of positive entropy is spanned by three Bernoulli factors.

Introduction.The results announced here are concerned with the isomorphism theory of dynamical systems.Many dynamical systems occur in nature, and although they are by definition deterministic processes, they often exhibit (what seems to be) random behavior to the observer.One of the major motivations of ergodic theory is to display isomorphisms between deterministic processes and processes which are clearly random in nature, such as Bernoulli shifts or Markov processes.To have physical significance, these isomorphisms should be continuous in some sense, and the suitable concept turns out to be that of Unitary isomorphism, defined below.This research provides a basic item for the development of the above ideas, in showing that two irreducible Markov processes with the same information content per unit time (= entropy) are finitarily isomorphic.That is, there is really only one type of stationary random behavior.The adjective finitary means that the map of one sequence space to the other is almost surely continuous in the sense that a finite section of the image sequence is determined by a sufficiently long finite section of the original sequence; generally, the length required depends upon the particular sequence on hand.THEOREM 1. Irreducible finite memory Markov shifts on a finite or countable state space and of the same entropy and period are finitarily isomorphic.It is easy to see that without loss of generality we can assume that the Markov shifts are of memory one.Also, since a Markov shift of period d is a direct product of d points with a mixing Markov shift, the base being a finite or countable union of cylinder sets, we can assume mixing.The proof of Theorem 1 makes use of the following.LEMMA.Let {X n } be a stationary mixing Markov process of a finite or countable state space A = {a x a 2 • • • }.Then, there are two stationary processes {Y n }, {Z n } on finite or countable state spaces B = {b x b 2 b 3 • • • } and C = {c x c 2 c 3 • • •} with at least three states each, and of the same entropy as {X n },

A two-person bargaining problem is considered. It is shown that under four axioms that describe the behavior of players there is a unique solution to such a problem. The axioms and the solution presented are different from those suggested by Nash. Also, families of solutions which satisfy a more limited set of axioms and which are continuous are discussed. WE CONSIDER a two-person bargaining problem mathematically formulated as follows. To every two-person game we associate a pair (a, S), where a is a point in the plane and S is a subset of the plane. The pair (a, S) has the following intuitive interpretation: a = (a1, a2) where ai is the level of utility that player i receives if the two players do not cooperate with each other. Every point x = (x1, x2) e S represents levels of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they do cooperate. We are interested in finding an outcome in S which will be agreeable to both players. This problem was considered by Nash [3] and his classical result was that under certain axioms there is a unique solution. However, one of his axioms of independence of irrelevant alternatives came under criticism (see [2, p. 128]). In this paper we suggest an alternative axiom which leads to another unique solution. Also, it was called to our attention by the referee that experiments conducted by H. W. Crott [1] led to the solution implied by our axioms rather than to Nash's solution. We also consider the class of continuous solutions which are required to satisfy only the axioms of Nash which are usually accepted. We give examples of families of such solutions.

The notion of complexity for compact convex sets introduced by Billera and Bixby is considered. It is shown that for n ≥ 3 n \geq 3 there are sets in R n {R^n} of complexity n n . Also for n = 3 n = 3 the maximal complexity is 3.

An example is given of a family of singular probability measures on the unit interval which are supported on a set of fractional Hausdorff dimension but cannot be represented as Hausdorff measures.

In a paper by Schutzenberger and Marcus (1959) full and completable codes were defined and investigated for the case of finite codes. The present paper compares the cases of a finite and an infinite code. We introduce the space Ω of all infinite sequences of letters and define a product probability on it. We then show that in the finite case the properties of fullness and completability are equivalent to the assertion that the probability of the message set (Section 1) is positive. In the infinite case this is no longer true, and an example is given of a full code with a message set of zero probability.