Infinitely fine partitions of measure spaces, Vladikavkaz Mathematical Journal, 1 (1999), no. 3, 53-59.

Brief review:

This work was completed in 1993-1994 and was originally circulated as a preprint. Some of the results were included in author's Master Thesis.

This work was inspired by P.Loeb's paper (1972) where the notion of an infinitely fine partition was introduced. In this construction nonstandard analysis is used to study the structure of an arbitrary measure space.

The idealization principle in Nelson's approach or the saturation principle in Robinson's approach guarantees that in any standard measurable set we can find a hyperfinite measurable partition which is finer than any standard measurable partition. We call such a partition an infinitely fine partition  (ifp). Two elements of an ifp are said to be equivalent if they can be joined and the partition will stay infinitely fine. Two elements are equivalent if and only if they cannot be separated by a standard measurable set. A monad of an ifp is the union of all equivalent elements. The partition of the space into monads is exactly the external partition generated by all the standard measurable sets, and does not depend on the ifp.

We immediately obtain some characterization of measurable functions and nonatomic measures. A standard bounded function is measurable if and only if it is nearly constant on every monad. A standard measure is nonatomic if and only if its value on any element of an ifp is infinitesimal.

Among the elements of an ifp forming a monad, there is a distinguished one, we call it the central element of the monad. Any element of an ifp generates some standard zero-one measure which assigns 1 to any standard set containing this element. Equivalent elements generate identical zero-one measure, this measure takes value 1 on the central element of the corresponding monad and vanishes on all the other elements of the ifp.

Consideration of the correspondent Stone space reveals the origin of the central elements and gives several beautiful remarks on them. It is well known that the algebra of measurable sets is isomorphic to the algebra of the clopen subsets of the Stone space. This isomorphism transforms an ifp on the measurable space into an ifp on the algebra of the clopen sets, and monads go to monads. Moreover, it turns over that the monads of this new ifp are exactly the topological monads of the Stone space. Since Stone space is compact, each monad contains one standard point. The element of the ifp which contains this point is the central element of the monad. And the corresponding element of the original ifp is also central relative to the original ifp.

This gives us some interesting properties of central elements. Any standard measure takes infinitesimal values on noncentral elements of ifp. If the measure of some central element is standard, then the measure vanishes on all other elements of this monad. The measure of a central element is always greater than its standard part.

All this technique gives a short and natural proof of Sobczik-Hammer Decomposition Theorem, which states that any measure is a sum of a nonatomic measure and a countable series of zero-one measures with some coefficients. The same reasoning also gives the proof of Sobczik-Hammer Decomposition Theorem for the vector-measure case.

Another application of ifp technique is a short proof of Horn-Tarsky theorem on extendibility of a measure from a subalgebra to an algebra.

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