Lebesgue space

A measure space $(M,\mathfrak B, \mu)$ (where $M$ is a set, $\mathfrak B$ is a $\sigma$-algebra of subsets of $M$, called measurable sets, and $\mu$ is a measure defined on the measurable sets), isomorphic to the "standard model" , consisting of an interval $\Delta$ and an at most countable set of points $\alpha_i$ (in "extreme" cases this "model" may consists of just the interval $\Delta$ or of just the points $\alpha_i$) endowed with the following measure $\mathfrak m$: on $\Delta$ one takes the usual Lebesgue measure, and to each of the points $\alpha_i$ one ascribes a measure $\mathfrak(\alpha_i) = \mathfrak m_i$; the measure is assumed to be normalized, that is, $\mu(M) = \mathfrak m(\Delta) + \sum\mathfrak m_i = 1$. The "isomorphism" can be understood here in the strict sense or modulo $0$; 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 $0$). One can give a definition of a Lebesgue space in terms of "intrinsic" properties of the measure space $(M,\mathfrak B, \mu)$ (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 $\sigma$-algebra on a measure space $(\mathfrak B, \mu)$ is generated by some automorphism of a Lebesgue space $M$. Under a number of natural operations, from a Lebesgue space one again obtains a Lebesgue space. Thus, a subset $A$ of positive measure in a Lebesgue space $M$ is itself a Lebesgue space (its measurable subsets are assumed to be those that are measurable in $M$, and the measure is $\mu_A(X)=\mu(X) / \mu(A)$); 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)