Regularity properties of analytic subsets of Polish spaces

We study three regularity properties of analytic sets of real numbers: Lebesgue measurability, the property of Baire, and the Weak Continuum Hypothesis. The study of regularity properties of analytic sets is most naturally situated in Polish spaces, a topological generalization\ud of the real numbers. We introduce the Baire space, a Polish space homeomorphic to the set of irrational numbers, and prove several transfer theorems which allow us to study the regularity properties of\ud analytic subsets of arbitrary Polish spaces by analyzing the Baire space. It is shown by these methods that every analytic set of real numbers is Lebesgue Measurable, and that every analytic set of an arbitrary Polish space has the property of Baire, and satisfies the Weak Continuum Hypothesis