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equivalent definitions of analytic sets
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For a paved space $(X,\mathcal{F})$ the $\mathcal{F}$ -analytic sets can be defined as the projections of sets in $(\mathcal{F}\times\mathcal{K})_{\sigma\delta}$ onto $X$ , for compact paved spaces $(K,\mathcal{K})$ . There are, however, many other equivalent definitions, some of which we list here.
In conditions 2 and 3 of the following theorem, Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ is the collection of sequences of natural numbers together with the product topology. In conditions 5 and 6, $Y$ can be any uncountable Polish space. For example, we may take $Y=\mathbb{R}$ with the standard topology.
Theorem Let $(X,\mathcal{F})$ be a paved space such that $\mathcal{F}$ contains the empty set, and $A$ be a subset of $X$ . The following are equivalent.
- $A$ is $\mathcal{F}$ -analytic.
- There is a closed subset $S$ of $\mathcal{N}$ and $\theta\colon \mathbb{N}^2\to\mathcal{F}$ such that \begin{equation*} A=\bigcup_{s\in S}\bigcap_{n=1}^\infty \theta\left(n,s_n\right). \end{equation*}
- There is a closed subset $S$ of $\mathcal{N}$ and $\theta\colon \mathbb{N}\to\mathcal{F}$ such that \begin{equation*} A=\bigcup_{s\in S}\bigcap_{n=1}^\infty \theta\left(s_n\right). \end{equation*}
- $A$ is the result of a Souslin scheme on $\mathcal{F}$ .
- $A$ is the projection of a set in $(\mathcal{F}\times\mathcal{G})_{\sigma\delta}$ onto $X$ , where $\mathcal{G}$ is the collection of closed subsets of $Y$ .
- $A$ is the projection of a set in $(\mathcal{F}\times\mathcal{K})_{\sigma\delta}$ onto $X$ , where $\mathcal{K}$ is the collection of compact subsets of $Y$ .
For subsets of a measurable space, the following result gives a simple condition to be analytic. Again, the space $Y$ can be any uncountable Polish space, and its Borel $\sigma$ -algebra is denoted by $\mathcal{B}$ . In particular, this result shows that a subset of the real numbers is analytic if and only if it is the projection of a Borel set from $\mathbb{R}^2$ .
Theorem Let $(X,\mathcal{F})$ be a measurable space. For a subset $A$ of $X$ the following are equivalent.
- $A$ is $\mathcal{F}$ -analytic.
- $A$ is the projection of an $\mathcal{F}\otimes\mathcal{B}$ -measurable subset of $X\times Y$ onto $X$ .
We finally state some equivalent definitions of analytic subsets of a Polish space. Again, $\mathcal{N}$ denotes Baire space and $Y$ is any uncountable Polish space.
Theorem For a nonempty subset $A$ of a Polish space $X$ the following are equivalent.
- $A$ is $\mathcal{F}$ -analytic.
- $A$ is the projection of a closed subset of $X\times\mathcal{N}$ onto $X$ .
- $A$ is the projection of a Borel subset of $X\times Y$ onto $X$ .
- $A$ is the image of a continuous function $f\colon Z\to X$ for some Polish space $Z$ .
- $A$ is the image of a continuous function $f\colon \mathcal{N}\to X$ .
- $A$ is the image of a Borel measurable function $f\colon Y\to X$ .
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"equivalent definitions of analytic sets" is owned by gel.
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| Keywords: |
analytic set, Polish space, Baire space, measurable space, paved space |
This object's parent.
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Cross-references: Borel measurable function, continuous function, Borel subset, Borel set, real numbers, analytic, measurable space, compact subsets, Souslin scheme, closed subset, the following are equivalent, empty set, standard topology, Polish space, uncountable, product topology, natural numbers, sequences, collection, Baire space, paved space
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This is version 3 of equivalent definitions of analytic sets, born on 2009-02-08, modified 2009-02-10.
Object id is 11609, canonical name is EquivalentDefinitionsOfAnalyticSets.
Accessed 411 times total.
Classification:
| AMS MSC: | 28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets) |
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Pending Errata and Addenda
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