Uniform open hulls for Borel sections of small measure

For every Borel $B\subseteq 2^{\omega }\times 2^{\omega }$, the function $x\mapsto \mu (B_x)$ is Borel. Let $\mathcal{D}$ be the class of Borel sets with this property. It contains every cylinder rectangle, since its section measure is a constant times a cylinder indicator. It contains the whole product, is closed under complements by $\mu ((B^c{)}_x)=1-\mu (B_x)$, and under countable disjoint unions by countable additivity and pointwise limits of partial sums. The cylinder rectangles form a $\pi$-system generating the product Borel algebra, so the elementary $\pi$-$\lambda$ (monotone-class) argument gives every Borel $B$. Explicitly, for a fixed rectangle the class of sets whose intersections with it lie in $\mathcal{D}$ is a Dynkin class; applying the same closure twice extends the assertion from rectangles to their generated sigma algebra.

Use the fixed length-lexicographic cylinder enumeration to code an open set by the set of cylinders listed in its union. Countable unions of open codes are Borel operations on codes: a cylinder belongs to the output list exactly when it appears in one of the input lists. Selecting one code from a countable list by a Borel integer-valued map is Borel as well.

We prove the stronger hull assertion simultaneously at all countable Borel ranks. If $H$ is open in the product, write it as the union of all basic product rectangles contained in it. This is a fixed countable enumeration; for each $x$, retain exactly the second-factor cylinders of rectangles whose first factor contains $x$. These are Borel coordinate tests and code the open section $H_x$ itself, with zero excess.

If $H={\bigcup }_i{H}_i$ and hull-code operators have been constructed for the $H_i$ at smaller rank, apply them with errors $ϵ2^{-i-2}$ and union their open sections. This union covers $H_x$; the points added outside $H_x$ lie in the union of the individual excess sets, whose total measure is at most ${\sum }_iϵ2^{-i-2}<ϵ$. The output code depends Borelly on $x$.

It remains to handle the complement stage of the Borel hierarchy. In its standard additive/multiplicative normal form, write the set under consideration as $A={\bigcap }_i{A}_i$ where $A_i$ decrease and belong to an additive class already handled at this stage of the Borel hierarchy. For ${\mathrm{\Pi }}_1^0$ these are decreasing open neighborhoods; at higher multiplicative ranks the usual normal form is a countable intersection of lower-rank additive sets. Finite intersections make the sequence decreasing. Use the induction hypothesis to obtain open $O_i(x)\supseteq (A_i{)}_x$ with $\mu (O_i(x)\setminus (A_i{)}_x)<ϵ\mathrm{/}2$. Put $K_i=\{x:\mu (((A_i\setminus A{)}_x))<ϵ\mathrm{/}2\}\mathrm{.}$ Each $K_i$ is Borel by step 1.1. Since $(A_i{)}_x$ decreases to $A_x$ and $\mu$ is finite, continuity from above makes the $K_i$ increasing with union all parameters. The least $i=i(x)$ with $x\in K_i$ is therefore a Borel integer-valued function. Set $O_x=O_{i(x)}(x)$. Then $\mu (O_x\setminus A_x)\le \mu (O_{i(x)}(x)\setminus (A_{i(x)}{)}_x)+\mu (((A_{i(x)}\setminus A{)}_x))<ϵ\mathrm{.}$ The selected open code is Borel by step 1.2.

Every Borel set has a well-founded countable construction code from open sets using complement and countable union. The two-part transfinite Borel-rank induction—additive classes first by countable unions of earlier multiplicative classes, then multiplicative classes by step 2.3—using steps 2.1, 2.2 and 2.3 gives the asserted code operator for the particular $H$; no pointwise arbitrary choice of hulls is made. If $\mu (H_x)=0$, the disjoint decomposition $O_x=H_x\cup (O_x\setminus H_x)$ gives $\mu (O_x)<ϵ$. ∎