# direct images of analytic sets are analytic

For measurable spaces^{} $(X,\mathcal{F})$ and $(Y,\mathcal{G})$, consider a measurable function^{} $f:X\to Y$. By definition, the inverse image ${f}^{-1}(A)$ will be in $\mathcal{F}$ whenever $A$ is in $\mathcal{G}$.
However, the situation is more complicated for direct images^{} (http://planetmath.org/DirectImage), which in general do not preserve measurability. However, as stated by the following theorem, the class of analytic subsets of Polish spaces^{} is closed under direct images.

###### Theorem.

Let $f\mathrm{:}X\mathrm{\to}Y$ be a Borel measurable function between Polish spaces $X$ and $Y$. Then, the direct image $f\mathit{}\mathrm{(}A\mathrm{)}$ is analytic whenever $A$ is an analytic subset of $X$.

Title | direct images of analytic sets are analytic |
---|---|

Canonical name | DirectImagesOfAnalyticSetsAreAnalytic |

Date of creation | 2013-03-22 18:46:33 |

Last modified on | 2013-03-22 18:46:33 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A05 |