Definition of Lebesgue Space

Question

I came across two definitions of Lebesgue space (also known as standard probability space) and I would like to know if they are equivalent. It seems to me so, but I think the second one can allow atoms.

(Sinai): $(M,\mathcal{A},\mu)$ is called Lebesgue space if $M$ is isomorphic mod 0 with the segment $[0, 1]$ carrying the standard Lebesgue measure.

From Sinai, Ya. G. (1994), Topics in ergodic theory.

(Aaronson): A Lebesgue space is a complete measure space wich is isomorphic to the completion of a standard measure space.

From Aaronson, J. (1997), An introduction to Infinite Ergodic Theory.

Furthermore, Sinai requires that $(M,\mathcal{A},\mu)$ be a measure space without points of positive measure.

$(X,\mathcal{B},\nu)$ is standard measure space if $(X,\mathcal{B})$ is a polish space equipped whith its collection of Borel sets.

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It seems to me that when Aaronson requires the space to be complete and isomorphic to a Polish space, he ends up avoiding atoms, but I'm not sure. Are these definitions equivalent? And the role of the measure in the Aaronson's definition was also not very clear.

Answer 1

In Topics, Ya Sinai only considers continuous Lebesgue spaces.

In the book Fomin, S. V., Cornfeld, I. P., Sinai, Y. G., Ergodic Theory, Springer Velag, 1982, Appendix 1, Lebesgue spaces are introduced following the ideas that V. A. Rohlin developed and presented in all their glory in Rohlin, V. A., On the fundamental ideas of measure theory, M. Sbornik, vol 25, number 67, pp. 107-150, 1949 (in Russian). Translated by AMS in 1952, Number 71.

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The notion of Lebesgue space introduced by Rohlin (and adopted throughout dynamical systems and in probability) is outlined below:

Definition: Suppose $(M,\mathscr{G},\mu)$ is a probability space. A countable collection of sets $\mathscr{B}=\{B_n:n\in\mathbb{N}\}\subset\mathscr{G}$ is a a basis for $(M,\mathscr{G},\mu)$ if:

• (i) For any $A\in \mathscr{G}$, there is $B\in\sigma(\mathscr{B})$ such that $A\subset B$ and $\mu(B)=\mu(A)$.
• (ii) For every distinct points $x,y\in M$, there exists $B\in\mathscr{B}$ such that either $x\in B$ and $y\in M\setminus B$, or $y\in B$ and $y\in M\setminus B$.

(i) says that $\mathscr{G}=\overline{\sigma(\mathscr{B})}^\mu=\{G\subset M: \exists E,F\in\sigma(\mathscr{B}),\, E\subset G\subset F,\, \mu(E)=\mu(F)\}$, the $\mu$-completion of $\sigma(\mathscr{B})$.

For any sequence $e:\mathbb{N}\rightarrow\{0,1\}$ define $\beta_e:\mathscr{B}\rightarrow\mathscr{G}$ as $\beta_e(B_n)=B_n$ if $e_n=0$ and $\beta_e(B_n)=M\setminus B_n$ if $e_n=1$. Part (ii) of the definition above implies that for any $e$, $\bigcap_n e(B_n)$ contains at the most one point.

Definition: The space $(M,\mathscr{G},\mu)$ is complete with respect to the basis $\mathscr{B}$ iff for any sequence $e\in \{0,1\}^{\mathbb{N}}$, $$ \bigcap_n\beta_e(B_n)\neq\emptyset$$

Completeness means implies that all singletons $\{x\}$, $x\in M$, are measurable and that every point $x\in M$ can be uniquely characterize by a sequence of $e\in\{0,1\}^{\mathbb{N}}$ through elements of the basis $\mathscr{B}$.

Definition: The space $(M,\mathscr{G},\mu)$ is calle complete $\mod 0$ with respect to a basis $\mathscr{B}$ iff there is a complete space $(\overline{M},\overline{\mathscr{G}},\overline{\mu})$ with respect to a base $\overline{B}$ such that $M\subset \overline{M}$, $B_n=\overline{B}_n\cap M$, and $\overline{\mu}^*(M)=1$, where $\overline{\mu}^*$ is the outer measure associated to $(\overline{M},\overline{\mathscr{G}},\overline{\mu})$.

It is a fact that if $\mathscr{B}$ and $\mathscr{B}'$ are two basis for $(M,\mathscr{G}, \mu)$, if $(M,\mathscr{G}, \mu)$ is complete $\mod 0$ with respect to $\mathscr{B}$, then it is complete $\mod 0$ with respect to $\mathscr{B}'$. Now the

finally we have:

Definition: A space $(M,\mathscr{G},\mu)$ that is complete $\mod 0$ with respect to a basis $\mathscr{B}$ is said to be a Lebesgue space, and the measure $\mu$ a Lebesgue measure.

Now we introduce the notion of equivalency between measure spaces

Definition: Two measure spaces $(M,\mathscr{G},\mu)$ and $(N,\mathscr{H},\nu)$ are isomorphic if there is a bijective map $f:M\rightarrow N$ such that $f^{-1}(H)\in\mathscr{G}$ for all $H\mathscr{H}$ and $f(G)\in\mathscr{H}$ for all $G\in \mathscr{G}$, $\nu(H)=\mu(f^{-1}(H))$ and $\mu(G)=\nu(f(G))$ for all $G\in\mathscr{G}$ and $H\in\mathscr{H}$.

Definition: > Definition: Two measure spaces $(M,\mathscr{G},\mu)$ and $(N,\mathscr{H},\nu)$ are isomorphic $\mod 0$ if there sets $A\in\mathscr{G}$ and $B\in\mathscr{H}$ such that $\mu(A)=0=\nu(B)$, and the subspaces $(M\setminus A,\{G\setminus A:G\in\mathscr{G}\},\mu)$ and $(N\setminus B,\{H\setminus A: H\in\mathscr{H}\},\nu)$ are isomorphic.

Two important results on this topic are:

Proposition 1: Every Lebesgue space with a continuous meausure is isomorphic $\mod 0$ to the unit interval $([0,1],\mathscr{M}([0,1]),\lambda)$ where $\lambda$ is the Lebesgue measure on $[0,1]$ and $\mathscr{M}([0,1])=\overline{\mathcal{B}([0,1])}^\lambda$ (the completion of the Borel $\sigma$-algebra with respect to $\lambda$).

Proposition 2: A Lebesgue measure is isomorphic $\mod 0$ to a space consisting $(M,\mathscr{G},\mu)$ consisting of an interval of length $0\leq\lambda(I)\leq 1$ and countable set $P=\{p_n\}$ of points such that $\sum_n\mu(\{p\}_n)=1-\lambda(I)$. The points in $P$ are the atoms of $\mu$ (provided $\mu(\{p\})>0$ for $p\in P$).

A useful result is the following:

Theorem: Suppose $(X, d)$ is a complete separable metric space (A Polish space) equipped with the Borel $\sigma$-algebra $\mathcal{B}(X)$. If $\mu$ is a probability on $\mathcal{B}(X)$ and $\mathscr{M}(B)=\overline{\mathcal{B}(X)}^\mu$, then $(X,\mathscr{M}(X),\mu)$ is a Lebesgue space.

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Comments:

1. The Appendix in Fomin, et. al. is short and does not contained many details, though the overall presentation gives good description of what a Lebesgue space is and why one bothers.

2. Rohlin's paper is not an easy read, or at least I have not been able to digest since I learnt of it over 20 years ago.

3. Rohlin's work on the axiomatic development of Lebesgue spaces and canonical measure space started in 1940 and motivated in part by his conversations with Andrei Kolmogorov. In his 1949 paper, Vladimir Rohlin mentions that he learned of a different axiomatization by Paul Halmos and John von Neumann (1942). Rohlin's axiomatization turned out to be an exceptionally convenient refinement of von Neumann and Kolmogorov's.

4. A modern and short presentation of Lebesgue-Rohlin spaces can be found in Bogachev, V. I., Measure Theory vol II, Springer, 2007, pp. 280.