# countable unions and intersections of analytic sets are analytic

A property of analytic sets^{} (http://planetmath.org/AnalyticSet2) which makes them particularly suited to applications in measure theory is that, in common with $\sigma $-algebras (http://planetmath.org/SigmaAlgebra), they are closed under countable^{} unions and intersections^{}.

###### Theorem 1.

Let $\mathrm{(}X\mathrm{,}\mathrm{F}\mathrm{)}$ be a paved space and ${\mathrm{(}{A}_{n}\mathrm{)}}_{n\mathrm{\in}\mathrm{N}}$ be a sequence^{} of $\mathrm{F}$-analytic sets. Then, ${\mathrm{\bigcup}}_{n}{A}_{n}$ and ${\mathrm{\bigcap}}_{n}{A}_{n}$ are $\mathrm{F}$-analytic.

A consequence of this is that measurable sets^{} are analytic, as follows.

###### Corollary.

Let $\mathrm{F}$ be a nonempty paving on a set $X$ such that the complement (http://planetmath.org/Complement) of any $S\mathrm{\in}\mathrm{F}$ is a union of countably many sets in $\mathrm{F}$.

Then, every set $A$ in the $\sigma $-algebra generated by $\mathrm{F}$ is $\mathrm{F}$-analytic.

For example, every closed subset of a metric space $X$ is a union of countably many open sets. Therefore, the corollary shows that all Borel sets are analytic with respect to the open subsets of $X$.

That the corollary does indeed follow from Theorem 1 is a simple application of the monotone class theorem.
First, as the collection^{} $a(\mathcal{F})$ of $\mathcal{F}$-analytic sets is closed under countable unions and finite intersections, it will contain all finite unions of finite intersections of sets in $\mathcal{F}$ and their complements, which is an algebra (http://planetmath.org/RingOfSets). Then, Theorem 1 says that $a(\mathcal{F})$ is closed under taking limits of increasing and decreasing sequences of sets. So, by the monotone class theorem, it contains the $\sigma $-algebra generated by $\mathcal{F}$.

Title | countable unions and intersections of analytic sets are analytic |
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Canonical name | CountableUnionsAndIntersectionsOfAnalyticSetsAreAnalytic |

Date of creation | 2013-03-22 18:45:18 |

Last modified on | 2013-03-22 18:45:18 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A05 |