Lebesgue space

A measure space (where is a set, is a -algebra of subsets of , called measurable sets, and is a measure defined on the measurable sets), isomorphic to the "standard model" , consisting of an interval and an at most countable set of points (in "extreme" cases this "model" may consists of just the interval or of just the points ) endowed with the following measure : on one takes the usual Lebesgue measure, and to each of the points one ascribes a measure ; the measure is assumed to be normalized, that is, . The "isomorphism" can be understood here in the strict sense or modulo ; one obtains, respectively, a narrower or wider version of the concept of a Lebesgue space (in the latter case one can talk about a Lebesgue space modulo ). One can give a definition of a Lebesgue space in terms of "intrinsic" properties of the measure space (see [1]–[3]).

A Lebesgue space is the most frequently occurring type of space with a normalized measure, since any complete separable metric space with a normalized measure (defined on its Borel subsets and then completed in the usual way) is a Lebesgue space. Apart from properties common to all measure spaces, a Lebesgue space has a number of specific "good" properties. For example, any automorphism of a Boolean -algebra on a measure space is generated by some automorphism of a Lebesgue space . Under a number of natural operations, from a Lebesgue space one again obtains a Lebesgue space. Thus, a subset of positive measure in a Lebesgue space is itself a Lebesgue space (its measurable subsets are assumed to be those that are measurable in , and the measure is ); the direct product of finitely or countably many Lebesgue spaces is a Lebesgue space. Other properties of Lebesgue spaces are connected with measurable partitions (cf. Measurable decomposition).

References

[1]	P.R. Halmos, J. von Neumann, "Operator methods in classical mechanics. II" Ann. of Math. , 43 : 2 (1942) pp. 332–350
[2]	V.A. Rokhlin, "On mean notions of measure theory" Mat. Sb. , 25 : 1 (1949) pp. 107–150 (In Russian)
[3]	J. Haezendonck, "Abstract Lebesgue–Rokhlin spaces" Bull. Soc. Math. Belg. , 25 : 3 (1973) pp. 243–258

Comments

Cf. also [a1] for a discussion of Lebesgue spaces and measurable partitions, including an intrinsic description of Lebesgue spaces.

References

[a1]	I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) pp. Appendix 1 (Translated from Russian)