Measurable decomposition

measurable partition, of a measure space $ ( M, \mu ) $

A partition (cf. Decomposition) $ \xi $ of the space into disjoint subsets (called the elements of the partition) that can be obtained as the partition into level sets of some measurable function (with numerical values) on $ M $. This definition can be restated in terms of "intrinsic" properties of the partition (see [1]). In accordance with the general tendency in questions of measure theory to ignore sets of measure zero, a measurable partition is often understood to be a partition measurable modulo 0, that is, equivalent modulo 0 to some measurable partition (two partitions $ \xi $ and $ \eta $ of a measure space $ M $ are equivalent modulo 0 if there exists a set $ N $ of measure 0 such that the partitions of $ M \setminus N $ consisting of the intersections with $ M \setminus N $ of the elements of $ \xi $ and $ \eta $ coincide).

Although the definition given makes sense for any measure space, in fact measurable partitions are almost always considered for Lebesgue spaces (cf. Lebesgue space) (and sometimes for spaces having the properties of the latter to some extent, for example, for spaces with perfect measures; see [2] and [3] and Perfect measure), since in these spaces measurable partitions have a number of "good" properties. Thus, in this case there exists a system of conditional measures (or, as used to be said in earlier times [1], a canonical system of measures) belonging to measurable partitions. This is a system of measures $ \mu ( \cdot \mid C) $ on the elements $ C $ of the partition $ \xi $ which enables one to consider integration with respect to $ \mu $ as repeated integration: first, integration is performed over the $ C $ and with respect to appropriate $ \mu ( \cdot \mid C) $, and then it is necessary to integrate the result, which can be looked upon as a function on the quotient space $ M/ \xi $, with respect to the natural measure $ \mu _ \xi $ on the latter space (by definition, $ M/ \xi $ has the elements of $ \xi $ as points, while its measurable subsets are those with measurable pre-images under the natural projection $ \pi : M \rightarrow M/ \xi $; the measure is defined to be $ \mu _ \xi ( A) = \mu ( \pi ^ {-} 1 ( A)) $). Interpreting $ ( M, \mu ) $ as a space of elementary events in probability theory, one can say that the system of conditional measures is an "improvement" of the conditional probability, closely connected with the specific Lebesgue space; for arbitrary spaces of elementary events, the conditional probability cannot, in general, be interpreted using some set of measures on the elements of some partition.

Non-measurable (and non-measurable modulo 0) partitions are by no means always "pathological" objects, like non-measurable sets or functions. For example, the partition of the phase space of an ergodic dynamical system into its trajectories can have a completely "classical" origin; it is simply that its properties are different from those of a measurable partition.

References

[1]	V.A. Rokhlin, "On mean notions of measure theory" Mat. Sb. , 25 : 1 (1949) pp. 107–150 (In Russian)
[2]	B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)
[3]	V.V. Sazonov, "On perfect measures" Izv. Akad. Nauk. SSSR Ser. Mat. , 26 : 3 (1962) pp. 391–414 (In Russian)

Comments

A useful result in ergodic theory concerning Lebesgue spaces and measurable partitions is the Rokhlin–Halmos theorem. In its strong form it states: Let $ \pi = ( A _ {1} \dots A _ {r} ) $ be a finite measurable partition of a Lebesgue space $ ( M, {\mathcal A} , \mu ) $ and let $ T $ be an automorphism of this space. Then, for any $ \epsilon > 0 $ and any $ n \in \mathbf N $ there exists an $ E \in {\mathcal A} $ such that $ E, TE \dots T ^ {n-} 1 E $ are pairwise disjoint, $ \mu ( \cup _ {i=} 0 ^ {n-} 1 T ^ {i} E) > 1- \epsilon $ and $ \mu ( E \cap A _ {j} ) = \mu ( E) \mu ( A _ {j} ) $, $ j = 1 \dots r $. (The weak form results by taking the trivial partition $ ( M) $.)

References

[a1]	I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Chapt. 10, §5 (Translated from Russian)