[proofplan] The strategy is the standard Maz'ya / Federer-Ziemer route, structured around a single black-box covering lemma. The lemma asserts that vanishing $p$-capacity forces the existence, for every $\eta > 0$, of a countable cover of $E$ by balls $\{B(x_k, r_k)\}$ whose radii satisfy a polynomial sum bound $\sum_k r_k^{n-p} \le C(n,p)\, \eta$ in the subcritical range $1 \le p < n$, or a logarithmic sum bound $\sum_k (\log(R_0/r_k))^{-(n-1)} \le C(n)\, \eta$ in the borderline $p = n$. We treat both forms of this lemma as established results in nonlinear potential theory: full proofs are in Maz'ya, *Sobolev Spaces in Mathematical Physics* (2nd ed., Springer, 2011); Heinonen-Kilpeläinen-Martio, *Nonlinear Potential Theory of Degenerate Elliptic Equations* (Oxford, 1993); Evans-Gariepy, *Measure Theory and Fine Properties of Functions* (CRC, 2nd ed., 2015); and Federer, *Geometric Measure Theory* (Springer, 1969). Granted this lemma, the rest of the proof is an algebraic split. Crucially, we do *not* refine the cover: refining a radius-$r$ ball into smaller radius-$\rho$ balls preserves $\sigma$-content only when $\sigma = n$, so naive subdivision breaks at exponent $\sigma = n - p < n$. Instead, given a fixed Hausdorff scale $\delta > 0$, we partition the cover into balls of radius $\le \delta/2$ (used as-is, contributing $(2 r_k)^s$) and balls of radius $> \delta/2$ (covered each by $\lesssim (r_k/\delta)^n$ balls of radius $\delta/2$, contributing $(r_k/\delta)^n \delta^s$ in aggregate). To avoid technical headaches with cutoff functions, we state and prove the theorem first for *bounded* $E$, where admissible test functions $\varphi$ may be chosen directly with support in any fixed neighbourhood of $E$. The polynomial split $r_k^s$ vs.\ $r_k^{n-p}$ in case 1 and $r_k^n = r_k^{n-p} \cdot r_k^p \le R_0^p\, r_k^{n-p}$ in case 2 both reduce to multiples of $\sum r_k^{n-p} \le C(n,p)\eta$. Sending $\eta \to 0$ at fixed $\delta, R_0$ kills $\mathcal{H}^s_\delta(E)$, and $\delta$ was arbitrary, so $\mathcal{H}^s(E) = 0$ for bounded $E$. A countable union argument $E = \bigcup_j (E \cap B(0, j))$ — using monotone subadditivity of $p$-capacity to lift capacity-zero from $E$ to each piece $E \cap B(0, j)$, and countable subadditivity of Hausdorff measure to lift the conclusion back from the pieces to $E$ — removes the boundedness. The borderline $p = n$ uses the same two-case skeleton: large balls are covered by $\delta/2$-balls and small balls are kept, with the algebraic input the inequality $r^s \le C \cdot (\log(R_0/r))^{-(n-1)}$ valid uniformly on $r \in (0, R_0/e]$. [/proofplan] [step:Setup, capacity, Hausdorff content, and reduction to bounded $E$] Variational $p$-capacity. For $1 \le p \le n$ and a Borel set $E \subseteq \mathbb{R}^n$, \begin{align*} \operatorname{Cap}_p(E) := \inf\left\{ \int_{\mathbb{R}^n} |\nabla \varphi|^p \, d\mathcal{L}^n : \varphi \in C_c^\infty(\mathbb{R}^n),\ \varphi \ge 1 \text{ on a neighbourhood of } E \right\}. \end{align*} This functional is monotone (if $E_1 \subseteq E_2$ then $\operatorname{Cap}_p(E_1) \le \operatorname{Cap}_p(E_2)$) and countably subadditive on Borel sets. The standard truncation $\min(\varphi, 1)^+$ shows the admissible class may be restricted to $0 \le \varphi \le 1$. Hausdorff $s$-content and measure. The pre-measure at scale $\delta$ is \begin{align*} \mathcal{H}^s_\delta(E) := \inf\left\{ \sum_k (\operatorname{diam} A_k)^s : E \subseteq \bigcup_k A_k,\ \operatorname{diam} A_k \le \delta \right\}, \end{align*} and $\mathcal{H}^s(E) = \lim_{\delta \to 0^+} \mathcal{H}^s_\delta(E) = \sup_{\delta > 0} \mathcal{H}^s_\delta(E)$. Equivalence of the spherical and net variants up to a dimensional constant lets us work with ball covers throughout, and proving the pre-measure vanishes for every $\delta > 0$ is equivalent to proving the full Hausdorff measure vanishes. Hausdorff measure is countably subadditive: $\mathcal{H}^s(\bigcup_j E_j) \le \sum_j \mathcal{H}^s(E_j)$. Notation. We use the average notation $\fint_{B} \varphi := \mathcal{L}^n(B)^{-1} \int_B \varphi \, d\mathcal{L}^n$, where $\mathcal{L}^n(B(x, r)) = \omega_n r^n$ and $\omega_n$ denotes the Lebesgue volume of the unit ball in $\mathbb{R}^n$. Hausdorff dimension is $\dim_{\mathcal{H}}(E) := \inf\{ s \ge 0 : \mathcal{H}^s(E) = 0 \}$. Reduction to bounded $E$. We prove the theorem first under the additional assumption that $E$ is bounded. The general case follows by a countable union: given an arbitrary Borel $E \subseteq \mathbb{R}^n$ with $\operatorname{Cap}_p(E) = 0$, decompose \begin{align*} E = \bigcup_{j \ge 1} E_j, \qquad E_j := E \cap B(0, j), \end{align*} a countable union of bounded Borel sets. Each $E_j \subseteq E$, so by monotonicity of $p$-capacity $\operatorname{Cap}_p(E_j) \le \operatorname{Cap}_p(E) = 0$, hence $\operatorname{Cap}_p(E_j) = 0$. The bounded case (proved below) yields $\mathcal{H}^s(E_j) = 0$ for every $j$, and countable subadditivity of $\mathcal{H}^s$ gives \begin{align*} \mathcal{H}^s(E) \le \sum_{j \ge 1} \mathcal{H}^s(E_j) = 0. \end{align*} The same union argument lifts $\mathcal{H}^s(E_j) = 0$ for all $s > 0$ (the borderline $p = n$ conclusion) to $\mathcal{H}^s(E) = 0$ for all $s > 0$, equivalently $\dim_{\mathcal{H}}(E) = 0$. Henceforth fix bounded $E \subseteq \mathbb{R}^n$ with $\operatorname{Cap}_p(E) = 0$, and choose $R \ge 1$ with $E \subseteq B(0, R)$. Set the support pocket $R_0 := R + 1$. By the definition of $\operatorname{Cap}_p(E)$ as an infimum and outer regularity, for every $\eta > 0$ we may choose an admissible test function \begin{align*} \varphi \in C_c^\infty(B(0, R_0)), \qquad \varphi \ge 1 \text{ on a neighbourhood of } E, \qquad 0 \le \varphi \le 1, \end{align*} with \begin{align*} \int_{\mathbb{R}^n} |\nabla \varphi|^p \, d\mathcal{L}^n < \eta. \end{align*} Indeed, the unrestricted infimum equals zero, so for any $\eta > 0$ there exists an admissible $\psi \in C_c^\infty(\mathbb{R}^n)$ with $\int |\nabla \psi|^p < \eta/2^{p}$ (using truncation to take $0 \le \psi \le 1$). The variational $p$-capacity admits an equivalent characterization for bounded sets where the test functions are required *a priori* to have support in any fixed open neighbourhood of $\overline{E}$; this bounded-domain capacity coincides with the unrestricted one up to a dimensional multiplicative factor (Maz'ya, *Sobolev Spaces in Mathematical Physics*, 2nd ed., 2011, Section 2.6, Theorem 2.6.2 on the equivalence of variational capacities). So the bounded-domain infimum is also zero, and we may directly choose $\varphi$ supported in $B(0, R_0)$ with $\int |\nabla \varphi|^p < \eta$. Fix also $s > n - p$ (or $s > 0$ when $p = n$) and $\delta > 0$. The goal is $\mathcal{H}^s_\delta(E) = 0$. The subcritical case $1 \le p < n$ is handled in Steps 2-4; the borderline $p = n$ in Steps 5-6. [/step] [step:Capacity-Hausdorff covering lemma (subcritical case)] The technical engine is the following covering bound, a standard result in nonlinear potential theory which we state and use as a black box. [claim:Capacity-Hausdorff Covering Lemma Subcritical] Let $1 \le p < n$. There is a constant $C(n, p) > 0$ such that for every $\varphi \in C_c^\infty(\mathbb{R}^n)$ with $\varphi \ge 1$ on an open neighbourhood of a Borel set $E \subseteq \mathbb{R}^n$, and every reference scale $R_0 > 0$ such that $\operatorname{supp} \varphi \subseteq B(0, R_0)$ and $E \subseteq B(0, R_0)$, there exists a countable cover $E \subseteq \bigcup_k B(x_k, r_k)$ with $r_k \le R_0$ for every $k$ and \begin{align*} \sum_k r_k^{n - p} \le C(n, p) \int_{\mathbb{R}^n} |\nabla \varphi|^p \, d\mathcal{L}^n. \tag{$\ast$} \end{align*} This is a standard capacity-Hausdorff comparison result in nonlinear potential theory. References: Maz'ya, *Sobolev Spaces in Mathematical Physics*, 2nd ed. (Springer, 2011), Theorem 11.6.1, p.\ 595 (capacitary inequality for Hausdorff content in the subcritical range); Federer, *Geometric Measure Theory* (Springer, 1969), Section 2.10.19 (capacity-Hausdorff measure inequality); Evans-Gariepy, *Measure Theory and Fine Properties of Functions*, 2nd ed.\ (CRC Press, 2015), Section 4.7.2 (Hausdorff dimension and Sobolev capacity). [/claim] Applying the lemma with our chosen $\varphi$ (with $\int |\nabla \varphi|^p < \eta$) and reference scale $R_0$, we obtain a cover $E \subseteq \bigcup_k B(x_k, r_k)$ with $r_k \le R_0$ for every $k$ and \begin{align*} \sum_k r_k^{n - p} \le C(n, p)\, \eta. \end{align*} [/step] [step:Two-case algebraic split at scale $\delta$] We must convert $(\ast)$ into a Hausdorff $s$-content bound at scale $\delta$. We do *not* refine the cover. Refinement would multiply the $(n-p)$-content by an unbounded factor: covering a radius-$r$ ball by $\sim (r/\rho)^n$ radius-$\rho$ balls inflates the $\sigma$-content from $r^\sigma$ to $\sim (r/\rho)^n \rho^\sigma = r^\sigma (r/\rho)^{n-\sigma}$, and for $\sigma = n - p < n$ this factor $(r/\rho)^p > 1$ is not bounded by a dimensional constant. Instead we partition the cover at the threshold $\delta/2$ and treat the two pieces by separate algebraic estimates. Partition. Set \begin{align*} K_{\text{small}} := \{k : r_k \le \delta/2\}, \qquad K_{\text{large}} := \{k : r_k > \delta/2\}. \end{align*} For $k \in K_{\text{small}}$ keep the ball; the cover-element $B(x_k, r_k)$ has diameter $2 r_k \le \delta$, admissible for $\mathcal{H}^s_\delta$, and contributes $(2 r_k)^s$. For $k \in K_{\text{large}}$ cover $B(x_k, r_k)$ by balls of radius $\delta/2$. The standard packing observation — a ball of radius $r$ in $\mathbb{R}^n$ admits a cover by $N$ balls of radius $\rho \le r$ with \begin{align*} N \le N_{\text{pack}}(n) (r/\rho)^n, \end{align*} where $N_{\text{pack}}(n)$ is a dimensional volume-comparison constant — applied with $\rho = \delta/2$ gives $N_k \le N_{\text{pack}}(n) (2 r_k / \delta)^n$ balls of radius $\delta/2$. Each such ball has diameter $\delta$ and is admissible for $\mathcal{H}^s_\delta$, contributing $\delta^s$. Summing both contributions, \begin{align*} \mathcal{H}^s_\delta(E) \le \sum_{k \in K_{\text{small}}} (2 r_k)^s + \sum_{k \in K_{\text{large}}} N_{\text{pack}}(n) \left( \frac{2 r_k}{\delta} \right)^n \delta^s = 2^s \!\!\sum_{k \in K_{\text{small}}}\!\! r_k^s + N_{\text{pack}}(n)\, 2^n\, \delta^{s - n} \!\!\sum_{k \in K_{\text{large}}}\!\! r_k^n. \tag{$\heartsuit$} \end{align*} We now bound each of the two sums on the right by a multiple of $\sum_k r_k^{n-p} \le C(n, p)\, \eta$, the algebraic split being different in each case. Small balls. For $k \in K_{\text{small}}$ we have $r_k \le \delta/2$ and $s - (n - p) > 0$, hence $r_k^{s - (n - p)} \le (\delta/2)^{s - (n - p)}$ and \begin{align*} \sum_{k \in K_{\text{small}}} r_k^s = \sum_{k \in K_{\text{small}}} r_k^{s - (n - p)} \cdot r_k^{n - p} \le (\delta/2)^{s - (n - p)} \sum_{k \in K_{\text{small}}} r_k^{n - p} \le (\delta/2)^{s - (n - p)} C(n, p)\, \eta. \end{align*} Large balls. For $k \in K_{\text{large}}$ we use $r_k \le R_0$ (the lemma's a-priori bound) to write $r_k^n = r_k^p \cdot r_k^{n - p} \le R_0^p \cdot r_k^{n - p}$, hence \begin{align*} \sum_{k \in K_{\text{large}}} r_k^n \le R_0^p \sum_{k \in K_{\text{large}}} r_k^{n - p} \le R_0^p \cdot C(n, p)\, \eta. \end{align*} [/step] [step:Bound on Hausdorff content and the $\eta \to 0$ limit] Combining the small- and large-ball estimates from Step 3 with ($\heartsuit$), \begin{align*} \mathcal{H}^s_\delta(E) &\le 2^s (\delta/2)^{s - (n - p)} C(n, p)\, \eta + N_{\text{pack}}(n)\, 2^n\, \delta^{s - n}\, R_0^p\, C(n, p)\, \eta \\ &= \Big[ 2^{n - p} \delta^{s - (n - p)} + N_{\text{pack}}(n)\, 2^n\, \delta^{s - n}\, R_0^p \Big] \cdot C(n, p)\, \eta \\ &=: C(n, p, s, \delta, R_0)\, \eta. \end{align*} The bracketed factor depends on $n, p, s, \delta, R_0$ but not on the cover and not on the test function $\varphi$. Since $E$ has zero $p$-capacity, $\eta > 0$ was arbitrary; letting $\eta \to 0^+$ at fixed $n, p, s, \delta, R_0$ gives \begin{align*} \mathcal{H}^s_\delta(E) = 0. \end{align*} This holds for every $\delta > 0$, so $\mathcal{H}^s(E) = \sup_{\delta > 0} \mathcal{H}^s_\delta(E) = 0$. This proves the conclusion in the subcritical range $1 \le p < n$ for every $s > n - p$, for bounded $E$. The countable union argument from Step 1 (decompose general $E = \bigcup_j (E \cap B(0, j))$, apply the bounded case to each piece, sum via countable subadditivity of $\mathcal{H}^s$) extends the conclusion to arbitrary Borel $E$. [/step] [step:The borderline case $p = n$, capacity lemma] For $p = n$ the conclusion is sharper: $\operatorname{Cap}_n(E) = 0$ implies $\mathcal{H}^s(E) = 0$ for every $s > 0$, equivalently $\dim_{\mathcal{H}}(E) = 0$. The Sobolev embedding $W^{1, n} \hookrightarrow L^\infty$ fails, so polynomial ball-capacity estimates degenerate and are replaced by the borderline (logarithmic) estimate. Fix in advance the reference scale $R_0' := 2 R_0$, depending only on the support pocket $R_0$ from Step 1. [claim:Capacity-Hausdorff Covering Lemma Borderline] Let $p = n$. There is a constant $C(n) > 0$ such that for every $\varphi \in C_c^\infty(\mathbb{R}^n)$ with $\varphi \ge 1$ on an open neighbourhood of a Borel set $E \subseteq \mathbb{R}^n$, and every reference scale $R_0' > 0$ such that $\operatorname{supp} \varphi \subseteq B(0, R_0)$ with $R_0' \ge 2 R_0$ and $E \subseteq B(0, R_0)$, there exists a countable cover $E \subseteq \bigcup_k B(x_k, r_k)$ with $r_k \le R_0'/e$ for every $k$ and \begin{align*} \sum_k \frac{1}{(\log(R_0' / r_k))^{n - 1}} \le C(n) \int_{\mathbb{R}^n} |\nabla \varphi|^n \, d\mathcal{L}^n. \tag{$\dagger$} \end{align*} This is a standard logarithmic-capacity result in nonlinear potential theory at the critical Sobolev exponent. References: Heinonen-Kilpeläinen-Martio, *Nonlinear Potential Theory of Degenerate Elliptic Equations* (Oxford University Press, 1993), Section 2.26 (logarithmic capacity bounds for $W^{1, n}$ and capacity-Hausdorff comparison at the borderline); Maz'ya, *Sobolev Spaces in Mathematical Physics*, 2nd ed.\ (Springer, 2011), Section 13.2 (capacitary criteria at the critical exponent and logarithmic Hausdorff content). [/claim] Applying the lemma with our chosen $\varphi$ (with $\int |\nabla \varphi|^n < \eta$) and reference scale $R_0' = 2 R_0$, we obtain a cover $E \subseteq \bigcup_k B(x_k, r_k)$ with $r_k \le R_0'/e$ for every $k$ and \begin{align*} \sum_k \frac{1}{(\log(R_0' / r_k))^{n - 1}} \le C(n)\, \eta. \tag{$\dagger'$} \end{align*} [/step] [step:Two-case algebraic split (borderline) and conclusion] Fix $s > 0$. We need an algebraic input that converts the logarithmic sum ($\dagger'$) into a polynomial $s$-content sum. The function \begin{align*} g(r) := r^s \cdot (\log(R_0' / r))^{n - 1}, \qquad r \in (0, R_0'/e], \end{align*} is bounded. Indeed, set $u := \log(R_0' / r)$, so $r = R_0' e^{-u}$ and $u \in [1, \infty)$ corresponds to $r \in (0, R_0'/e]$. Then $g = (R_0')^s e^{-s u} u^{n - 1}$, a continuous function of $u \in [1, \infty)$ that equals $(R_0'/e)^s$ at $u = 1$ and tends to $0$ as $u \to \infty$ (since $e^{-su}$ decays faster than $u^{n-1}$ grows, for any $s > 0$). Hence $g$ attains its supremum on $[1, \infty)$ and \begin{align*} M := M(n, s, R_0') := \sup_{r \in (0, R_0'/e]} r^s (\log(R_0' / r))^{n - 1} < \infty. \end{align*} For every $r \in (0, R_0'/e]$, \begin{align*} r^s \le M \cdot (\log(R_0' / r))^{-(n - 1)}. \tag{$\flat$} \end{align*} Since $r_k \le R_0'/e$ for every $k$ in the cover from Step 5, ($\flat$) applies uniformly across $k$. Two-case split at scale $\delta$. We do not refine the cover; the borderline case fails the same naive-refinement test as the subcritical case (the volume-comparison factor $(r_k/\rho)^n$ is not absorbed by the logarithmic ratio $(\log(R_0'/r_k))^{n-1} / (\log(R_0'/\rho))^{n-1}$ into a dimension-only constant). Partition \begin{align*} K_{\text{small}} := \{k : r_k \le \delta/2\}, \qquad K_{\text{large}} := \{k : r_k > \delta/2\}. \end{align*} For $k \in K_{\text{small}}$, keep the ball ($\operatorname{diam} = 2 r_k \le \delta$), contributing $(2 r_k)^s$. For $k \in K_{\text{large}}$, cover by $N_k \le N_{\text{pack}}(n) (2 r_k / \delta)^n$ balls of radius $\delta/2$ (each diameter $\delta$, admissible), contributing $N_k \delta^s$. Hence \begin{align*} \mathcal{H}^s_\delta(E) \le 2^s \sum_{k \in K_{\text{small}}} r_k^s + N_{\text{pack}}(n)\, 2^n\, \delta^{s - n} \sum_{k \in K_{\text{large}}} r_k^n. \end{align*} Small balls. By ($\flat$) and ($\dagger'$), \begin{align*} \sum_{k \in K_{\text{small}}} r_k^s \le M \sum_{k \in K_{\text{small}}} (\log(R_0'/r_k))^{-(n-1)} \le M \cdot C(n)\, \eta. \end{align*} Large balls. For $k \in K_{\text{large}}$ we have $\delta/2 < r_k \le R_0'/e$, in particular the radii lie in a bounded interval. The map $r \mapsto r^n \cdot (\log(R_0'/r))^{n-1}$ is continuous and strictly positive on the closed interval $[\delta/2, R_0'/e]$, hence attains its supremum; let \begin{align*} M_n := M_n(n, \delta, R_0') := \sup_{r \in [\delta/2, R_0'/e]} r^n (\log(R_0'/r))^{n - 1} < \infty. \end{align*} (For $\delta/2 \ge R_0'/e$, the set $K_{\text{large}}$ is empty and this contribution is zero; we may assume $\delta/2 < R_0'/e$, taking $\delta$ sufficiently small.) Then for $k \in K_{\text{large}}$, \begin{align*} r_k^n \le M_n \cdot (\log(R_0'/r_k))^{-(n-1)}, \end{align*} and summing, \begin{align*} \sum_{k \in K_{\text{large}}} r_k^n \le M_n \sum_{k \in K_{\text{large}}} (\log(R_0'/r_k))^{-(n-1)} \le M_n \cdot C(n)\, \eta. \end{align*} Conclusion. Combining, \begin{align*} \mathcal{H}^s_\delta(E) \le \big[ 2^s M + N_{\text{pack}}(n)\, 2^n\, \delta^{s - n}\, M_n \big] \cdot C(n)\, \eta =: C(n, s, \delta, R_0')\, \eta. \end{align*} The bracketed factor depends on $n, s, \delta, R_0'$ but not on the cover and not on $\varphi$. Letting $\eta \to 0^+$ at fixed $n, s, \delta, R_0'$ gives $\mathcal{H}^s_\delta(E) = 0$. This holds for every $\delta > 0$, so $\mathcal{H}^s(E) = 0$. Since $s > 0$ was arbitrary, $\dim_{\mathcal{H}}(E) = 0$, for bounded $E$. The countable union argument from Step 1 extends the conclusion to arbitrary Borel $E$. This completes the proof in both ranges. [/step]