Shadows of the axiom of choice in the universe $L(\mathbb {R})$

Abstract

We show that several theorems about Polish spaces, which depend on the axiom of choice ($\mathcal {AC}$), have interesting corollaries that are theorems of the theory $\mathcal {ZF} + \mathcal {DC}$, where $\mathcal {DC}$ is the axiom of dependent choices. Surprisingly it is natural to use the full $\mathcal {AC}$ to prove the existence of these proofs; in fact we do not even know the proofs in $\mathcal {ZF} + \mathcal {DC}$. Let $\mathcal {AD}$ denote the axiom of determinacy. We show also, in the theory $\mathcal {ZF} + \mathcal {AD} + V = L(\mathbb {R})$, a theorem which strenghtens and generalizes the theorem of Drinfeld (Funct Anal Appl 18:245–246, 1985) and Margulis (Monatshefte Math 90:233–235, 1980) about the unicity of Lebesgue’s measure. This generalization is false in $\mathcal {ZFC}$, but assuming the existence of large enough cardinals it is true in the model $\langle L(\mathbb {R}),\in \rangle$.