Boldface Sigma-one-three measurability

Work in Cantor space $\mathcal{C}=2^{\mathbb{N}}$, with auxiliary real quantifiers ranging over Baire space $\mathcal{N}={\mathbb{N}}^{\mathbb{N}}$ as in Cantor and Baire sequence spaces and coordinate codings. Fix a real parameter $x$.

A set $A\subseteq \mathcal{C}$ is ${\mathrm{\Sigma }}_3^1(x)$ when membership in $A$ has the form

$a\in A⟺\mathrm{\exists }y\mathrm{\forall }z\mathrm{\exists }w\theta (a,y,z,w,x),$

where $y,z,w$ range over Baire space and $\theta$ is arithmetic: all its quantifiers are number quantifiers and its atomic statements are those of a fixed recursive decoding of the sequences involved. Thus the leading real quantifiers are one existential, one universal and one existential, the last being the one that may be dummy. A set is boldface ${\mathrm{\Sigma }}_3^1$ when it is ${\mathrm{\Sigma }}_3^1(x)$ for some real $x$, and the regularity assertion boldface ${\mathrm{\Sigma }}_3^1$-measurability means that every such set belongs to the completed coin-measure domain specified below.

The measured domain. In applications assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The proof of Dyadic coding supplies coin measure and its completed Lebesgue transfer uses DC only in its step 1.2 to derive Countable Choice; after that step, its cylinder-pullback construction of the Borel coin probability $\nu$ uses only the resulting Countable Choice hypotheses. Thus the same construction is available directly under the present assumption. Write $\mathcal{B}$ for its Borel sigma-algebra. Define $\bar{\mathcal{B}}:=\{E\subseteq \mathcal{C}:E=B\cup N,B,Z\in \mathcal{B},N\subseteq Z,\nu (Z)=0\}\mathrm{.}$ For such a representation put $\bar{\nu }(E):=\nu (B)$. This is exactly the completion construction of The completion domain and proposed completed set function of a measure space, so Assuming countable choice, every measure space has a unique complete extension to its completion proves that this value is independent of the representation and is a complete measure on the displayed sigma-algebra. Thus the regularity assertion is precisely $\mathrm{\forall }x\in \mathcal{N}\mathrm{\forall }A\subseteq \mathcal{C}(A\in {\mathrm{\Sigma }}_3^1(x)⟹A\in \bar{\mathcal{B}})\mathrm{.}$ The dyadic lemma supplies the Borel measure; the completion theorem supplies its completed domain and measure. No completion or measure transport is inferred from a homeomorphism between sequence spaces.

Elementary codings. Coordinate pairing gives the homeomorphisms $\mathcal{C}\cong {\mathcal{C}}^{\mathbb{N}}$ and $\mathcal{N}\cong {\mathcal{N}}^{\mathbb{N}}$, so finite or countable tuples of the respective real codes may be folded into one code. The published map from $\mathcal{N}$ is a homeomorphism only onto the subspace $D\subseteq \mathcal{C}$ of sequences with infinitely many $1$s; no homeomorphism $\mathcal{N}\cong \mathcal{C}$ and no measure transport along that subspace map is asserted here. Adding a dummy final existential real quantifier shows that every ${\mathrm{\Sigma }}_2^1(x)$ subset of $\mathcal{C}$ is ${\mathrm{\Sigma }}_3^1(x)$: prefix a redundant $\mathrm{\exists }w$ and ignore $w$ in $\theta$. This inclusion is the one used below when a ${\mathrm{\Sigma }}_3^1$-measurability hypothesis is applied to the ${\mathrm{\Sigma }}_2^1(x)$ null-code order. The quantifier-prefix definition uses no choice. The measured interpretation above is used under Countable Choice, which licenses both the Borel coin-measure construction and its completion.