Bernoulli Scheme - Generalizations

Generalizations

Most of the properties of the Bernoulli scheme follow from the countable direct product, rather than from the finite base space. Thus, one may take the base space to be any standard probability space, and define the Bernoulli scheme as

This works because the countable direct product of a standard probability space is again a standard probability space.

As a further generalization, one may replace the in integers by a countable discrete group, so that

For this last case, the shift operator is replaced by the group action

for group elements and understood as a function (any direct product can be understood to be the set of functions, as this is the exponential object). The measure is taken as the Haar measure, which is invariant under the group action:

These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case.