Hausdorff Measure Formulation of Goldstern's Principle

• The paper demonstrates that unlike the Lebesgue case, the Hausdorff measure version fails in L for sets with dimension less than 1 even with countable vertical sections.
• It defines the Hausdorff measure using gauge functions and explains σ-ideals formed by null sets within metric spaces.
• Using a co-analytic scale and Martin-Löf randomness, the construction yields a union with full Hausdorff dimension, challenging traditional null-additivity.

The Hausdorff measure version of Goldstern’s principle characterizes the interplay between descriptive set theory, measure theory, and the structure of σ-ideals defined by Hausdorff measures, especially in relation to co-analytic (Π^1_1) monotone families. Central to this study is the discovery that, contrary to the Lebesgue measure case, the Hausdorff measure analogue fails in the constructible universe $L$ for sets of dimension less than 1, even when all monotone vertical sections are countable. This distinction has foundational implications for the understanding of small sets, dimension theory, and regularity properties of definable sets in the context of higher pointclasses.

1. Classical Goldstern’s Principle and Lebesgue Measure

Goldstern’s principle (GP) for Lebesgue measure asserts that for a boldface pointclass $\Gamma$, whenever a set $A \subseteq \omega^\omega \times 2^\omega$ is monotone in $x$ and each vertical section $A_x$ is $\mu$-null, the union $\bigcup_{x \in \omega^\omega} A_x$ must also be $\mu$-null. Formally,

GP($\Gamma$): If $A \subseteq \omega^\omega \times 2^\omega$ is in $\Gamma$, monotone in $x$, and each $A_x$ has Lebesgue measure zero, then $\mu(\bigcup_x A_x) = 0$.

Goldstern [Goldstern ’93] established that GP($\Pi^1_1$) holds for Lebesgue measure, employing measure-theoretic forcing and absoluteness arguments to guarantee that no co-analytic monotone family of null sets can cover a non-null set (Goto, 28 Dec 2025).

2. Hausdorff Measure: Definitions and Formulation

For metric spaces $(X, d)$, a gauge function $f: [0, \infty) \to [0, \infty)$ is a nondecreasing, right-continuous function with $f(0) = 0$. For $A \subseteq X$ and $\delta > 0$:

$$H^f_\delta(A) = \inf \left\{ \sum_{n < \omega} f(\mathrm{diam}(C_n)) : A \subseteq \bigcup_{n < \omega} C_n, \,\mathrm{diam}(C_n) \leq \delta \right\},$$

and the $f$-Hausdorff measure is $H^f(A) = \lim_{\delta \to 0} H^f_\delta(A)$. Specializing to $f(r) = r^s$ yields the $s$-dimensional Hausdorff measure $H^s$. The σ-ideal $N^f_X$ consists of sets with σ-finite $f$-Hausdorff measure; alternatively, “null” is defined as $H^f(A) = 0$.

The Hausdorff-analogue Goldstern’s principle is:

GP($\Gamma$, $I$): If $A \subseteq \omega^\omega \times X$ is in $\Gamma$, monotone in $x$, and each $A_x \in I$, then $\bigcup_x A_x \in I$.

The central question is whether GP($\Pi^1_1$, $N^f_{2^\omega}$) holds—i.e., does every monotone co-analytic family of $f$-Hausdorff null subsets yield a null union for the standard Cantor space $2^\omega$?

3. Failure of Hausdorff Measure GP($\Pi^1_1$) in $L$

Section 3.2 of Goto (Goto, 28 Dec 2025) demonstrates:

Theorem 3.6: In $V = L$, there exists a monotone $\Pi^1_1$ set $A \subseteq \omega^\omega \times 2^\omega$ with countable vertical sections $A_x$, yet the union $\bigcup_{x \in \omega^\omega} A_x$ achieves Hausdorff dimension 1. For all power-gauges $f(r) = r^s$ with $0 < s < 1$,

$$\text{GP}(\Pi^1_1, N^f_{2^\omega}) \text{ fails in } L.$$

This result crucially relies on the construction of a $\Pi^1_1$ scale and coding reals of full effective dimension, thereby circumventing the nullity of vertical sections through diagonalization and randomness, leading to a non-null union under Hausdorff measure.

4. Construction and Proof Techniques

The proof builds on several key elements:

• Slaman’s Dimension Lemma: There exists an infinite co-infinite recursive set $R \subseteq \omega$ such that for every Martin-Löf random $X$ over oracle $B$, any $X^*$ agreeing with $X$ outside $R$ preserves effective Hausdorff dimension 1 relative to $B$.
• $\Pi^1_1$-Scale and Randomness: A $\Pi^1_1$ scale $\{x_\alpha : \alpha < \omega_1\}$ in $\omega^\omega$ is constructed, and for each $\alpha$, a real $y_\alpha$ codes the $L$-least Martin-Löf random real $r_\alpha$ over $y_\alpha$. The real $r^*_\alpha$ is crafted so that $r^*_\alpha$ agrees with $r_\alpha$ outside $R$ but “decodes” $x_\alpha$ on $R$.
• Dimension Argument: By Slaman’s lemma, each $r^*_\alpha$ is of effective dimension 1, hence $S = \{r^*_\alpha : \alpha < \omega_1\}$ has classical Hausdorff dimension 1 by the Lutz–Lutz theorem.
• Set Definition: $$A = \{(x, y) : y \in S \ \text{and} \ \text{decode}(y) \leq^* x \}$$ yields countable vertical sections and co-analytic monotonicity, yet the union $S$ is large in Hausdorff dimension.

This construction exploits the combinatorial and definability properties available in $L$ to defeat the Hausdorff analogue of GP.

5. Dichotomy Between Lebesgue and Hausdorff Measure Cases

The success of GP($\Pi^1_1$) under Lebesgue measure is attributed to the power of measure-theoretic forcing: positive measure unions admit Borel witnesses and are subject to random real forcing, which, combined with monotonicity and $\omega^\omega$-bounding, precludes non-null unions. In the σ-finite setting, decomposition and measure-isomorphism reduce the problem to the core Lebesgue case.

The failure for fixed $s \in (0,1)$ dimensional Hausdorff measures stems from the ability to diagonalize across all $x$ while constructing a union of full dimension. The existence of co-analytic scales of length $\omega_1$ in $L$ and the padding of elements via Martin-Löf randomness ensure the effective dimension remains maximal in the union, subverting the null-additivity required by GP.

6. Corollaries, Extensions, and Open Problems

Several corollaries and further results emerge:

• For any continuous doubling gauge $f$ on a compact metric $X$,

$\text{GP}(\Pi^1_1, N^f_X) \text{ holds in ZFC},$

relying on similar forcing and bounding arguments as in the Lebesgue case, with decomposition into non–σ-finite and σ-finite components, and the Sion–Sjerve theorem for Borel submeasures.

• An open question (Problem 5.1) asks whether some forcing extension may realize

$$\text{GP(all, } N^f_{2^\omega}) \text{ for every continuous doubling gauge } f,$$

that is, for all definable sets and all Hausdorff measures simultaneously.

• Additional results address variants of GP($\Gamma$) for larger pointclasses, interval-partition ideals, and ramifications for small-set ideals (strong measure zero, null-additive). Notably:
  • Solovay’s measure-uniformization principle implies GP(all).
  • GP(all) yields $\operatorname{cov}(M) \neq \mathfrak{b}$, introducing new cardinal characteristics.
  • GP(all) equates the strong-measure-zero ideal to the null-additive ideal.
  • $\Sigma^1_2$-Lebesgue measurability entails GP($\Sigma^1_2$).

Collectively, these findings from Goto (Goto, 28 Dec 2025) underscore a pronounced divergence between Lebesgue and Hausdorff measures in the field of co-analytic sets, instigating several open avenues for GP($\Gamma, I$) concerning alternative ideals.