Talk:Krylov–Bogolyubov theorem

Horrible Colors
The colors in this article are horrible and make me want to gouge my eyes out rather than to read the theorems. Mcxz (talk) 20:24, 7 June 2011 (UTC)

Found on newsgroup
The following excerpt from sci.math may be of interest to future editors.  The Relationship Between Measure-Theoretic and Topological Notions:

It does not make sense to say that ergodicity is EQUIVALENT to TT (topological transitivity). A topological dynamical system, i.e. X compact metric with T: X \rightarrow X continuous, has a different structure from a measure theoretical dynamical system, i.e., (X, \M, \mu) a probability measure space with T:X \rightarrow X a measureable mapping preserving the \mu measure, in the sense that for all measureable sets E, \mu T^{-1}(E) = \mu (E).

However, given a TDS (topological dynamical system), there are always MTDS's (measure-theoretical dynamical systems) associated with it. First, take \M to be the Borel algebra (smallest sigma algebra containing all the open and closed sets). Then the Krylov-Bogoliubov Theorem says that every T continuous has at least one T-invariant measure, i.e. \mu such that (X, Borel, \mu, T) is a measure theoretical dynamical system. Moreover, the set of Borel measures on X form a compact convex subset of a separable Banach space, an infinite dimensional ``simplex'' whose vertices correspond to delta measures on X, and for each T, the set of T-invariant measures forms a ``sub-simplex''. (You can actually draw this for the case of X a finite space with the discrete topology; the geometry is quite beautiful, although rather misleading vis a vis infinite X.) A beautiful theorem says that the ergodic T-invariant measures are exactly the vertices of the sub-simplex of T-invariant measures! The point is this:
 * EVERY TDS is associated with at least one ergodic MTDS!

Also, TDS's clearly have ``a higher level of structure'' than MTDS's, so your assertion about TT and ergodicity confused two levels of structure.

[Note: A delta measure \delta_x is indeed a point mass: for all Borel measureable sets E, \delta_x(E) = 1 if x \in E and 0 otherwise.]

Nevertheless, there are relations between topological and measure theoretic conditions. For instance, if (X,T) is an invertible TDS (T is a homeo), and if there exists an ergodic T-invariant measure which gives positive measure to every nonempty open set, then (X, T) is TT.  Incidentally, the spelling “Bogoliubov” gets somewhat more Google hits. --KSmrqT 00:17, 27 October 2006 (UTC)

Noted. Krylov-Bogoliubov theorem redirects to the main article, Krylov-Bogolyubov theorem. Sullivan.t.j 10:07, 27 October 2006 (UTC)

Possible source to cite?
The Springer Online Encyclopædia of Mathematics has an article on the Krylov–Bogolyubov method of averaging which may be of interest. --KSmrqT 03:24, 27 October 2006 (UTC)

The section about Markov processes is poorly written
I say this for no other reason than that it is never stated that time t takes real values.

Since this is an encylopedia, there is no way that a well written article could possibly omit this piece of information.