Null and meagre master codes are cofinal

Suppose every $H_x$ is null. By [F1] and the countable selection permitted by [F4], for each $j$ obtain Borel open codes $O_j(x)\supseteq H_x$ with $\mu (O_j(x))<2^{-j-2}$. For an open-coded set, its canonical prefix-free cylinders are exactly the basic cylinders contained in the open set whose immediate parent is not contained in it (with the root handled separately). The predicate $[s]\subseteq O_j(x)$ is Borel in $x$: compactness of $[s]$ turns inclusion in the enumerated open union into existence of a finite subcover, a countable disjunction of finite code tests. These prefix-free cylinders partition $O_j(x)$ and their measures sum to $\mu (O_j(x))$. Enumerate all pairs $(j,s)$ in a fixed order, putting the corresponding cylinder at its slot when it is canonical and the empty set otherwise. Write this clopen sequence as $(D_k(x){)}_k$. It depends Borelly on $x$ and ${\sum }_k\mu (D_k(x))\le {\sum }_j{2}^{-j-2}<1$.

Suppose every $H_x$ is meagre. By [F2] obtain Borel closed nowhere-dense codes $F_j(x)$ covering it. Put $V_n(x)=2^{\omega }\setminus {\bigcup }_{j\le n}{F}_j(x)$; this is dense open. List all basic cylinders $U_s$ contained in $V_n(x)$, repeating a fixed cylinder if necessary to make an infinite sequence. Inclusion $U_s\subseteq V_n(x)$ is Borel in $x$: for each $j\le n$ it requires $U_s\subseteq 2^{\omega }\setminus F_j(x)$, equivalently a finite subcover of the compact cylinder $U_s$ by the cylinders in that coded open complement. This is a finite conjunction of countable disjunctions of finite code tests. The resulting Borel sequence of cylinder indices is a valid meagre code $g_x$ with $V_n(g_x)=V_n(x)$. As $H_x$ lies in the union of the $F_j(x)$, it lies in $M_{g_x}$.

Let $T_h(x)={\sum }_{k\ge h}\mu (D_k(x))$, a Borel pointwise limit of finite partial sums. Its value decreases to zero. Starting with $h_0=0$, choose $h_{n+1}>h_n$ as the least integer with $T_{h_{n+1}}(x)\le 2^{-(n+1)}$. The threshold tests are Borel, so each $h_n(x)$ is Borel. Define $E_n(x)={\bigcup }_{h_n(x)\le k<h_{n+1}(x)}{D}_k(x)$, a finite clopen union. Then $\mu (E_n(x))\le T_{h_n(x)}(x)\le 2^{-n}$. Its index in the fixed clopen enumeration can be chosen canonically by least search, hence Borelly. This gives a valid null code $f_x$. Every $y\in H_x$ belongs to at least one canonical cylinder from each $O_j(x)$; these have distinct pair-slots as $j$ varies, so $y$ belongs to infinitely many $E_n(x)$. Thus $H_x\subseteq N_{f_x}$.

Let $A\in {\mathcal{N}}_{\mathcal{C}}$. By the definition [F3] choose a Borel null set $B\supseteq A$. Apply [F1] to the constant Borel family $H=\mathcal{C}\times B$ and the fixed parameter $x_0=0^{\omega }$, at errors $2^{-j-2}$, to obtain open sets $O_j\supseteq B$ with $\mu (O_j)<2^{-j-2}$; [F4] permits choosing the sequence of hull codes. Their intersection $G$ is a Borel null hull of $A$. The one-parameter versions of steps 1.1 and 2.1 applied to $G$ produce a null master set containing $G$ and hence $A$. For an arbitrary $A\in {\mathcal{M}}_{\mathcal{C}}$, take its defining sequence $(A_j)$ of nowhere-dense sets. Each $\overline{A_j}$ is closed nowhere dense by [F5], so $B={\bigcup }_j\overline{A_j}$ is Borel and meagre and contains $A$. Apply step 1.2 to the constant Borel family $\mathcal{C}\times B$ and evaluate at $0^{\omega }$ to obtain a meagre master set containing $B$, hence $A$. Taking the closures is canonical and uses no additional choice. This proves cofinality and the uniform Borel clauses. ∎