analytic set
(Note: this entry concerns analytic sets^{} as used in measure theory. For the definition in analytic spaces see analytic set (http://planetmath.org/AnalyticSet)).
For a continuous map^{} of topological spaces^{} it is known that the preimages^{} of open sets are open, preimages of closed (http://planetmath.org/ClosedSet) sets are closed and preimages of Borel sets are themselves Borel measurable. The situation is more difficult for direct images^{} (http://planetmath.org/DirectImage). That is if $f:X\to Y$ is continuous^{} then it does not follow that $S\subseteq X$ being open/closed/measurable implies the same property for $f(S)$. In fact, $f(X)=\mathrm{Image}(f)$ need not even be measurable. One of the few things that can be said, however, is that $f(S)$ is compact^{} whenever $S$ is compact. Analytic sets are defined in order to be stable under direct images, and their theory relies on the stability of compact sets. This is a fruitful concept because, as it turns out, all measurable sets^{} are analytic^{} and all analytic sets are universally measurable.
A subset $S$ of a Polish space^{} $X$ is said to be analytic (or, a Suslin set) if it is the image of a continuous map $f:Z\to X$ from another Polish space $Z$ — see Cohn. It is then clear that $g(S)$ will again be analytic for any continuous map $g:X\to Y$ between Polish spaces. Indeed, $g(S)$ will be the image of $g\circ f$.
Here, we instead give the following definition which applies to arbitrary paved spaces and, in particular, to all measurable spaces^{}. Furthermore, for Polish spaces, it can be shown to be equivalent^{} to the definition just mentioned above.
Recall that for a paved space $(X,\mathcal{F})$, ${\mathcal{F}}_{\sigma \delta}$ represents the collection^{} of countable^{} intersections^{} of countable unions of elements of $\mathcal{F}$, and that a paving is compact if every subcollection satisfying the finite intersection property has nonempty intersection.
Definition.
Let $\mathrm{(}X\mathrm{,}\mathrm{F}\mathrm{)}$ be a paved space. Then a set $A\mathrm{\subseteq}X$ is said to be $\mathrm{F}$-analytic, or analytic with respect to $\mathrm{F}$, if there exists a compact paved space $\mathrm{(}K\mathrm{,}\mathrm{K}\mathrm{)}$ and an $S\mathrm{\in}{\mathrm{(}\mathrm{F}\mathrm{\times}\mathrm{K}\mathrm{)}}_{\sigma \mathit{}\delta}$ such that
$$A=\pi (S).$$ |
Here, $\mathrm{F}\mathrm{\times}\mathrm{K}$ is the product paving and $\pi \mathrm{:}X\mathrm{\times}K\mathrm{\to}X$ is the projection map $\pi \mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}x$.
Writing this out explicitly, there are doubly indexed sequences of sets ${A}_{m,n}\in \mathcal{F}$ and ${K}_{m,n}\in \mathcal{K}$ such that
$$S=\bigcap _{n=1}^{\mathrm{\infty}}\bigcup _{m=1}^{\mathrm{\infty}}{A}_{m,n}\times {K}_{m,n}.$$ |
and,
$$A=\{x\in X:(x,y)\in S\text{for some}y\in K\}.$$ |
Although this allows for $\mathcal{K}$ to be any compact paving it can be shown that it makes no difference^{} if it is just taken to be the collection of compact subsets of, for example, the real numbers.
For a measurable space $(X,\mathcal{F})$, a subset of $X$ is simply said to be analytic if it is $\mathcal{F}$-analytic, and a subset of a topological space is said to be analytic if it is analytic with respect to the Borel $\sigma $-algebra (http://planetmath.org/BorelSigmaAlgebra).
References
- 1 K. Bichteler, Stochastic integration with jumps. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.
- 2 Donald L. Cohn, Measure theory. Birkhäuser, 1980.
- 3 Claude Dellacherie, Paul-André Meyer, Probabilities and potential. North-Holland Mathematics Studies, 29. North-Holland Publishing Co., 1978.
- 4 Sheng-we He, Jia-gang Wang, Jia-an Yan,Semimartingale theory and stochastic calculus. Kexue Chubanshe (Science Press), CRC Press, 1992.
- 5 M.M. Rao, Measure theory and integration. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 265. Marcel Dekker Inc., 2004.
Title | analytic set |
Canonical name | AnalyticSet1 |
Date of creation | 2013-03-22 18:44:52 |
Last modified on | 2013-03-22 18:44:52 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 9 |
Author | gel (22282) |
Entry type | Definition |
Classification | msc 28A05 |
Synonym | Suslin set |
Related topic | PavedSpace |
Related topic | BorelSigmaAlgebra |
Related topic | UniversallyMeasurable |
Defines | Suslin set |
Defines | analytic with respect to |