of them to be heads, and then I don't care about what comes after that". This can be written as the set where is "don't care" and is "heads". The volume of this space is again (obviously!) one-half. The above is enough to build up a measure-preserving dynamical system, in its entirety. The sets of or occurring in the 'th place are called cylinder sets. The set of all possible intersections, unions and complements of the cylinder sets then form the Borel set defined above. In formal terms, the cylinder sets form the base for a topology on the space of all possible infinite-length coin-flips. The measure has all of the common-sense properties one might hope for: the measure of a cylinder set with in the 'th position, and in the 'th position is obviously 1/4, and so on. These common-sense properties persist for set-complement and set-union: everything except for and in locations and obviously has the volume of 3/4. All together, these form the axioms of a sigma-additive measure; measure-preserving dynamical systems always use sigma-additive measures. For coin flips, this measure is called the Bernoulli measure. For the coin-flip process, the time-evolution operator is the shift operator that says "throw away the first coin-flip, and keep the rest". Formally, if is a sequence of coin-flips, then . The measure is obviously shift-invariant: as long as we are talking about some set where the first coin-flip is the "don't care" value, then the volume does not change: . In order to avoid talking about the first coin-flip, it is easier to define as inserting a "don't care" value into the first position: . With this definition, one obviously has that with no constraints on . This is again an example of why is used in the formal definitions. The above development takes a random process, the Bernoulli process, and converts it to a measure-preserving dynamical system The same conversion (equivalence, isomorphism) can be applied to any stochastic process. Thus, an informal definition of ergodicity is that a sequence is ergodic if it visits all of ; such sequences are "typical" for the process. Another is that its statistical properties can be deduced from a single, sufficiently long, random sample of the process (thus uniformly sampling all of ), or that any collection of random samples from a process must represent the average statistical properties of the entire process (that is,