The result

A Hausdorff gauge is a continuous nondecreasing function \(h:[0,\infty)\to[0,\infty)\) with \(h(0)=0\) and \(h(r)>0\) for \(r>0\). We use diameters in the definition \[\mathcal H^h(E)=\lim_{\delta\downarrow0}\mathcal H^h_\delta(E),\qquad \mathcal H^h_\delta(E)=\inf\left\{\sum_j h(\operatorname{diam}U_j): E\subset\bigcup_jU_j,\quad\operatorname{diam}U_j\le\delta\right\},\] where covers are countable and their sets may be arbitrary.

The gauge is the full infimum in (61), with coefficients specified by the iterated integrals in (57). The exceptional event may depend on the fixed \(\kappa\); the gauge is independent of both the sample and the interval. In particular, the trace outside its initial point has a nonzero sigma-finite Hausdorff measure.

The upper argument measures the entire trace in a bounded region. For \(\Gamma=\gamma([0,\infty))\) and \(B_m=[-m,m]+i[0,m]\), Proposition 20 proves \[ \mathbb E\,\mathcal H^{h_\kappa}(\Gamma\cap B_m)<\infty \qquad(m\in\mathbb N). \tag{1}\] Thus all these bounded-box measures are finite on one event, including at real-boundary trace points. Positivity in Theorem 1 is asserted for the stated positive-time intervals.

Multipoint estimates and one canonical mass

The gauge construction needs a trace mass with moments of every order and a way to compare its future in a small region with one fixed disk model. We establish these inputs through a multipoint probability bound and a conformally restarting mass. Both are also useful independently. We first give a direct product-supermartingale proof of the fixed-order estimate needed in an arbitrary physical domain. For a conformal map \(J:\mathbb H\to D\), let \(R_D(x)\) denote full conformal radius: \(R_D(x)=|F'(0)|\) when \(F:\mathbb D\to D\) is conformal, \(\mathbb D\) is the unit disk, and \(F(0)=x\). Write \(H_t\) for the surviving half-plane domain after the trace up to time \(t\), and \(D_t=J(H_t)\). A point is alive at \(t\) when it belongs to \(D_t\); its evolving radius is \(R_{D_t}(x)\). Set \(\alpha=2-d\) and \(a=2/\kappa\). For distinct interior points \(x_1,\ldots,x_m\), choose caps satisfying \[0<b_i\le R_D(x_i),\qquad b_i\le |x_i-x_j|\quad(j\ne i).\] Proposition 5 proves that the probability that each point reaches its own radius target \(e_i>0\) while alive is at most \[ C_{m,a}\prod_{i=1}^m\min\{1,(e_i/b_i)^\alpha\}. \tag{2}\] The constant depends only on the fixed order and parameter. It is uniform in the domain, the initial angles, and the order and times of the hits. A completed point may be swallowed before another target is reached. No sharp growth estimate in \(m\) is claimed. Integrating the collision singularities in this bound gives all the compact moments used in the mass construction.

The second ingredient is a single adapted increasing measure process \(\mu_t\) on the open domain, starting at zero. In the half-plane, Proposition 8 constructs it with nonatomic terminal measure, local total-variation continuity, and increments carried by the corresponding trace segments. Every nontrivial time interval has positive increment mass by Proposition 11. To describe its conditional mean, let \(g_S\) be the Loewner map of the surviving half-plane domain \(H_S\), with driving point \(U_S\) in the capacity normalization. For \(x\in H_S\) put \(\theta_S(x)=\arg(g_S(x)-U_S)\) and \(G_S(x)=R_{H_S}(x)^{-\alpha}\sin^{8/\kappa-1}\theta_S(x)\); put \(G_S(x)=0\) outside \(H_S\). Its remaining conditional mean is \[\mathbb E[\mu_\infty-\mu_S\mid\mathcal F_S]=G_S\,\mathop{}\!\mathrm{d}A\] for every fixed stopping time \(S\), including \(S=\infty\) with \(G_\infty=0\). Here \(\mathop{}\!\mathrm{d}A\) is planar area. The equality holds as a conditional measure kernel, so it includes nonnegative tests known at \(S\).

Conformal covariance weights the measure by \(|J'|^d\). At each fixed finite stopping time, Proposition 10 identifies the actual future increment with the conformally transported canonical mass of the fresh driver, simultaneously at every later time. This permits comparison of the actual mass near a marked point with one disk model after stopping at a prescribed conformal radius. The lower bound also uses first-visit accounting: mass already accumulated on the trace is not charged again at a later geometric return.

Why the moments determine the gauge

To bias the curve toward a marked interior point, stop when that point’s conformal radius reaches a prescribed smaller value and reweight the stopped law by the ratio of its terminal and initial Green weights. Paths that fail to reach the radius level receive weight zero. Section 2 proves that the weight has mean one and that these finite stopped laws are consistent as the radius decreases. We call them the centered laws.

Place the marked point at the origin of a unit disk and measure mass in its radius-\(1/4\) subdisk. The inner time is the logarithmic radius clock \(\ell=(2a)^{-1}\log(R_0/R)\), where \(R\) is the marked point’s current conformal radius. Let \(L_\ell\) be the mass accumulated in that subdisk by inner time \(\ell\). Under the centered law, the marked point’s angle in the corresponding half-plane coordinates evolves as a diffusion with invariant density proportional to \(\sin^{4a}\theta\). We average the initial angle against this distribution and write \(\mathbb P_{\mathrm{mix}}^*,\mathbb E_{\mathrm{mix}}^*\) for this averaged centered law and expectation. Then \[L_\ell\uparrow L,\qquad 0<\mathbb E_{\mathrm{mix}}^*L^n<\infty\quad(n\ge1).\] Section 5 specifies these numbers by Wiener, angle and area integrals, and then by product-normal coordinates and ordinary differential equation limits. This deterministic infinite-limit prescription fixes the normalization of the moments without referring to an unspecified random measure.

For each order \(n\), retain more than three quarters of its moment at a finite level \(\ell_n\). Repeat dyadic thresholds \(A_k=2^k\) in a finite batch with multiplicities \(N_{nk}\) determined by the \(n\)th and \((n+1)\)st moments. Lemma 16 gives the complementary bounds \[\sum_kN_{nk}\mathbb P_{\mathrm{mix}}^*\{L\ge16A_k\} \le\frac{4^{-n}}{1-2^{-n}},\qquad \sum_kN_{nk}\mathbb P_{\mathrm{mix}}^*\{L_{\ell_n}\ge A_k\}\ge\frac{2^n}{8} \quad(n\ge3).\] The first sum is summable over batches. The second supplies many opportunities for a success. Tests along the actual curve are separated in conformal-radius time so that angle mixing gives a conditional lower bound after each preceding test. This uses their actual stopped pasts, not independence. One final cleanup scale in each batch pays for points missed by all of its tests.

Writing \(b_{ni}\) and \(r_{ni}\) for the resulting thresholds and radii, the gauge interpolates their costs by the quadratic expressions \(b_{ni}r_{ni}^d\max\{1,(r/r_{ni})^2\}\) and takes the infimum over every entry. The quadratic factor accounts for subdivision in the plane. Radii in different batches may interleave.

For the lower bound, a strict past-mass identity uses the conditional Green kernel to bound expected mass in overfull squares through centered probabilities that a mass threshold is exceeded before the marked point is reached. The upper tails make these expected losses summable over all batch entries. Positive mass therefore remains on a set that avoids all overfull squares after one common batch cutoff. The resulting finite nonatomic measure is bounded by every late quadratic expression. Its finite total mass supplies the bounds for the finitely many excluded entries, so taking the full infimum is legitimate. Summing this one density bound over arbitrary countable covers proves positivity.

For the upper bound, successful finite tests force terminal-mass candidates. A greedy selection makes their charging balls disjoint, and one actual mass in a larger bounded region pays for their enlarged covering balls. Only adapted test failures enter the finite change of law; terminal candidate membership is used afterward through inclusion. The cleanup scale covers every missed cell, including the bottom row at the real boundary. These measurable covers have diameters tending to zero and uniformly bounded expected cost. Their lower limiting cost and Fatou’s Lemma give (1).

Prior work and proof organization

The Green-potential approach to trace mass has its roots in natural parametrization. Lawler and Sheffield constructed natural parametrization initially for \(0<\kappa<4(7-\sqrt{33})\) (Lawler and Sheffield 2011, Theorem 3.1); Lawler and Zhou obtained the full range \(0<\kappa<8\) (Lawler and Zhou 2013, Theorem 1). Lawler and Rezaei constructed \(d\)-dimensional Minkowski content and identified it, up to normalization, with natural parametrization (Lawler and Rezaei 2015, Theorem 1.1 and following discussion). Our common-kernel construction proves the properties needed here locally; identifying its mass with those measures is unnecessary. Multipoint estimates and their higher-moment consequences also have substantial predecessors (Lawler and Werness 2013; Rezaei and Zhan 2017). Rezaei–Zhan subsequently proved existence, local Hölder continuity, and bounds up to constants for chordal multipoint Green functions at every finite order (Rezaei and Zhan 2018, Theorem 1.1). The domain-uniform bound in (2) also follows from Rezaei–Zhan (Rezaei and Zhan 2017, Theorem 1.1) and the Koebe estimates. We give a direct product-supermartingale proof with each point frozen after its own target hit.

The use of regions carrying large mass together with a fine residual cover appears in Rezaei (Rezaei 2018, sec. 3, equations (9)–(11)). Zhan (Zhan 2019b, Remark 4.5) gives a conditional covering sketch for zero \(d\)-dimensional Hausdorff measure. The present finite batches, actual restart identities and all-time boundary cleanup supply the covering proof for the specified gauge. The quadratic envelope is related to the planar regularization of Peres–Solomyak (Peres and Solomyak 2005, Lemma 1.2 and Section 5); the strict past-mass calculation uses the same remaining-potential mechanism as Zhan’s decomposition along the SLE curve (Zhan 2019a, Proposition 2.2 and equations (4.3)–(4.5)).

Holden and Yuan identify a positive fine-mesh variation, up to a deterministic constant, with natural parametrization (Holden and Yuan 2026, Theorems 1.7–1.8). In the half-plane their result includes positive-time intervals that touch the boundary. Its ordered time partitions differ from the arbitrary spatial covers used here. The moment prescription proves exactness for its own gauge; it asserts neither a log-log asymptotic nor identification of Hausdorff measure with natural parametrization.

A later companion (OpenAI 2026, Theorem 1.1) gives exactness for the regular closed-form gauge \[r^d\bigl(\log\log(1/r)\bigr)^{(2-d)/2}\] at small radii, by quantitative mass tails and dense visits. Its argument establishes its own stopped-law, mass and covering estimates. The present argument leaves the asymptotic relation between that formula and the moment-defined gauge \(h_\kappa\) undetermined.

Section 2 proves finite centered laws, mixing and the remaining-potential identity in the required normalization. Sections 3 and 4 establish (2), the common mass kernel and actual restart. Section 5 recovers the deterministic disk moments; Section 6 turns them into batches and the gauge. Section 7 proves the strict past-mass comparison and arbitrary-cover lower bound. Section 8 constructs the all-time upper covers and assembles the theorem. The external geometric inputs are continuous trace generation (Rohde and Schramm 2005, Theorem 5.1) and the classical Koebe estimates (Beliaev 2015, sec. 3.2). Every subsequent centered-law, multipoint, mass and covering argument is proved below.

The local martingale and finite stopped laws

The finite-radius Green tilt and its centered angular diffusion are standard in the study of two-sided radial SLE; see Lawler–Zhou (Lawler and Zhou 2013, sec. 2.1) and Lawler–Rezaei (Lawler and Rezaei 2015, sec. 4.2, equations (38)–(40)). We give the cutoff argument and uniform mixing proof in the present full-conformal-radius normalization, with all changes of law made at finite stopped levels.

Fix \(x\in D\), put \(z=J^{-1}(x)\), and let \(T_x\) be its lifetime. Until \(T_x\), write \[Z_t=X_t+iY_t=g_t(z)-B_t,\qquad \theta_t=\arg Z_t,\qquad V_t=\frac{Y_t}{|Z_t|^2},\qquad s_x(t)=\int_0^t V_u^2\,\mathop{}\!\mathrm{d}u.\] Set \(R_t(x)=R_{D_t}(x)\). Conformal covariance of radius gives \[R_t(x)=\frac{2Y_t|J'(z)|}{|g_t'(z)|}.\] Since \(\mathop{}\!\mathrm{d}Z_t=aZ_t^{-1}\mathop{}\!\mathrm{d}t-\mathop{}\!\mathrm{d}B_t\), differentiation of this formula and Itô’s Formula for \(\log Z_t\) give \[\begin{align*} \mathop{}\!\mathrm{d}\log R_t(x)&=-2aV_t^2\,\mathop{}\!\mathrm{d}t, &R_t(x)&=R_0(x)e^{-2as_x(t)},\tag{9}\\ \mathop{}\!\mathrm{d}\theta_t&=V_t\,\mathop{}\!\mathrm{d}B_t+(1-2a)\cot\theta_t\,V_t^2\,\mathop{}\!\mathrm{d}t. \tag{10}\end{align*}\] Thus the angle in inner time has generator \[\mathcal L=\frac12\partial_\theta^2 +(1-2a)\cot\theta\,\partial_\theta.\] For every real \(r\), direct differentiation yields \[ \frac{\mathcal L\sin^r\theta}{\sin^r\theta} =-\frac r2+\frac{r(r-p)}2\cot^2\theta. \tag{11}\] Consequently \[ G_t(x)=R_t(x)^{-\alpha}\sin^p\theta_t(x) \quad (x\in D_t),\qquad G_t(x)=0\quad(x\notin D_t) \tag{12}\] is a local martingale during survival, with stochastic differential \(\mathop{}\!\mathrm{d}G_t=G_t p\cot\theta_t V_t\,\mathop{}\!\mathrm{d}B_t\).

For \(s\ge0\), let \(\tau_s(x)\) be the time at which the radius first attains \(R_0(x)e^{-2as}\) while \(x\) is alive; it is infinity if this never occurs. This is the inverse inner clock on successful paths.

Uniform mixing and the remaining potential

The stationary density and uniform mixing are also described in (Lawler and Rezaei 2015, sec. 4.2). The orthogonal-polynomial proof has a direct counterpart in Zhan (Zhan 2019b, Proposition 4.2). Writing \(X\) for the diffusion in that proposition, the substitution \(\theta_s=X_{4s/\kappa}/2\) gives our centered diffusion. We prove mixing uniformly over the initial angle and give the finite-lifetime estimates needed for the subsequent potential identity.

Controlling the interaction of two active points

Let \(\sigma_i\) and \(T_i\) be the first times when point \(i\) reaches radius \(b_i\) and \(e_i\), respectively, while alive, with value infinity if the corresponding radius is never reached. The point is active on \([\sigma_i,T_i)\) while it remains alive. Put \[U_i=\frac1{2a}\log\frac{b_i}{e_i},\qquad u_i(t)=\int_{\sigma_i}^t V_i(v)^2\,\mathop{}\!\mathrm{d}v\] on its active interval. Thus \(R_t(x_i)=b_i e^{-2au_i(t)}\), and it finishes when \(u_i=U_i\). We first take \(e_i<b_i\); constraints with \(e_i=b_i\) may be dropped.

For two surviving points put \[L_{ij}(t)=\log\left| \frac{Z_i(t)-\overline{Z_j(t)}}{Z_i(t)-Z_j(t)}\right|\ge0.\] The common Brownian driver cancels in the two differences. Consequently \[ \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}L_{ij} =\operatorname{Re}\left(-\frac a{Z_i\overline{Z_j}} +\frac a{Z_iZ_j}\right) =-2aV_iV_j. \tag{25}\] This is also a conformal invariant: for a disk uniformization \(F:\mathbb D\to D_t\) with \(F(0)=x_i\) and \(F(w)=x_j\), it equals \(-\log|w|\). Write \(\rho=|w|\) and \(L=-\log\rho\). The growth estimate (7) gives \[\frac{|x_i-x_j|}{R_t(x_i)} \le\frac{\rho}{(1-\rho)^2} =\frac1{4\sinh^2(L/2)}.\] During joint activity, the separation assumption and \(R_t(x_i)=b_i e^{-2au_i}\) imply \[L_{ij}\le2\operatorname{arsinh}\left( \frac12\sqrt{\frac{R_t(x_i)}{|x_i-x_j|}}\right) \le e^{-au_i}.\] Interchanging the two points proves \[ L_{ij}(t)\le e^{-a\max(u_i(t),u_j(t))} \qquad\text{during joint activity}. \tag{26}\] In particular, for \(c\ge0\) with \(2c<a\), \[ \int_{\text{joint activity}} e^{c(u_i+u_j)}V_iV_j\,\mathop{}\!\mathrm{d}t \le I(a,c):=\frac{e^{2c}}{2a(1-e^{2c-a})}. \tag{27}\] To see this, joint activity is an interval, and \(v=\max(u_i,u_j)\) increases on it. On its portion with \(k\le v<k+1\), the total decrease available to \(L_{ij}\) is at most \(e^{-ak}\), by (26). Equation (25) bounds the unweighted integral there by \(e^{-ak}/(2a)\), while the weight is at most \(e^{2c(k+1)}\). Summing over \(k\ge0\) proves (27), also for any truncated part of the joint active interval. This argument is made in \(D_t\) itself; no bound on the derivative of the initial map \(J\) enters it.

Regularized factors and the product supermartingale

The factor \(\sin^p\theta\) in the one-point martingale has no positive lower bound uniform in the angle at a target hit. We add a positive regularizing term at both ends of each active interval, so activation and completion have angle-uniform bounds. The added term decays away from those ends: this keeps its accumulated drift bounded independently of the interval length. The symmetric distance \(\min\{u,U_i-u\}\) records exactly that choice.

The exponent \(q\) will supply negative angular drift. Young’s Inequality will introduce a regularization loss with exponent \(M\), and we choose the decay rate \(\beta\) so that the pair interaction can absorb that loss. Use the explicit values \[ q=\tfrac12\min(1,p),\qquad \eta=\frac{1-q}{2-q}\in(0,\tfrac12),\qquad M=\frac{1+\eta}{1-2\eta}=\frac3q-2,\qquad \beta=\min\left(1,\frac a{4M}\right). \tag{28}\] Thus \(0<q<\min(1,p)\) and \(2M\beta\le a/2<a\). For an active point define \[ \delta_i(u)=e^{-\beta\min(u,U_i-u)},\qquad f_i(u,\theta)=\sin^p\theta+\delta_i(u)(1+\sin^q\theta),\qquad D_i=\frac{\delta_i}{f_i}\sin^{q-2}\theta_i. \tag{29}\] The factor for point \(i\) is \(1\) before activation, \[Q_i(t)=\frac13 e^{pu_i(t)/2}f_i(u_i(t),\theta_i(t))\] during activity, and the constant \(e^{pU_i/2}/3\) after completion. The activation jump is downward because \(f_i(0,\theta)\le3\); the completion jump is downward because \(f_i(U_i,\theta)\ge1\). After completion the point’s factor is frozen, including if that point is swallowed later. Figure 1 illustrates the separate endpoints.

Put \(Q=\prod_{i=1}^m Q_i\). If all points finish, this product has the deterministic value \[Q=3^{-m}\prod_i(b_i/e_i)^\alpha.\] Thus a uniform bound on the expectation of a discounted product will give the desired probability estimate, provided the total discount is deterministically bounded. The individual negative drifts absorb the singular angular terms, and the pair interaction estimate pays for what remains. We give both estimates explicitly. Write \(s=\sin\theta\), \(k=q(p-q)/2\), and \(\sigma=\delta'/\delta\), so \(|\sigma|\le\beta\le1\) almost everywhere. Formula (11) gives \[ \frac{(\partial_u+\mathcal L+p/2)f}{f} =\frac\delta f\left[ \frac p2+\frac{(p-q)(1+q)}2s^q-ks^{q-2} +\sigma(1+s^q)\right]. \tag{30}\] Let \[A=\frac p2+\frac{(p-q)(1+q)}2+2, \qquad h_0=\left(\frac{k}{2A}\right)^{1/(2-q)}\in(0,1).\] The bracket in (30) is at most \(A-ks^{q-2}\). For \(s<h_0\), adding \((k/2)D\) leaves a nonpositive expression. For \(s\ge h_0\), use \(f\ge s^p\ge h_0^p\). We obtain \[ \frac{(\partial_u+\mathcal L+p/2)f}{f} \le C_1\delta-c_1D, \qquad C_1=Ah_0^{-p},\quad c_1=k/2>0. \tag{31}\] All these constants depend only on \(a\). The corner of \(\delta\) at \(U_i/2\) causes no additional term: it is an absolutely continuous function of the finite-variation clock \(u_i\).

Because \(q<p\), \(q<1\), and \(0<\delta\le1\), angular differentiation gives \(|\partial_\theta f|\le(p+q)s^{q-1}\). Also \(\delta\le f\le3\), and \(\eta(q-2)=q-1\). Therefore \[ \frac{|\partial_\theta f|}{f} \le(p+q)\delta^{-1}s^{q-1} \le(p+q)3^\eta\delta^{-1-\eta}D^\eta. \tag{32}\] This bound includes the singular angular behavior when \(p<1\).

On active intervals, the normalized Brownian coefficient of \(Q_i\) is \((\partial_\theta f_i/f_i)V_i\). Itô’s Product Formula thus gives one cross term for each unordered active pair. Its absolute value is bounded, by (32), by \[C_2(\delta_i\delta_j)^{-1-\eta} (D_iV_i^2)^\eta(D_jV_j^2)^\eta(V_iV_j)^{1-2\eta}.\] Weighted Young’s Inequality with weights \(\eta,\eta,1-2\eta\) shows that for any \(\epsilon>0\) this is at most \[ \epsilon(D_iV_i^2+D_jV_j^2) +C_\epsilon(\delta_i\delta_j)^{-M}V_iV_j. \tag{33}\] Indeed, putting \(r=1-2\eta\), the weighted arithmetic–geometric mean inequality gives the displayed bound with \(C_\epsilon=rC_2^{1/r}(\eta/\epsilon)^{2\eta/r}\), and raises \(\delta_i\delta_j\) to exponent \(-(1+\eta)/r=-M\). For \(m>1\) choose \(\epsilon=c_1/[2(m-1)]\); for \(m=1\) there are no cross terms. Summing (31) and (33) absorbs the pair contributions into the negative terms. A nonnegative upper bound for the remaining normalized physical-time drift is \[k_t=C_1\sum_{i\text{ active}}\delta_iV_i^2 +C_3\sum_{\substack{i<j\\i,j\text{ active}}} (\delta_i\delta_j)^{-M}V_iV_j,\] where \(C_3\) depends only on \(m,a\) (and the pair sum is zero for \(m=1\)). Its total integral has a deterministic bound. Namely, \[\int_0^{U_i}\delta_i(u)\,\mathop{}\!\mathrm{d}u =\frac2\beta(1-e^{-\beta U_i/2})\le\frac2\beta, \qquad (\delta_i\delta_j)^{-M}\le e^{M\beta(u_i+u_j)}.\] Equation (27), with \(c=M\beta\), yields \[ \int k_t\,\mathop{}\!\mathrm{d}t\le K_{m,a}:=\frac{2mC_1}{\beta} +C_3\binom m2 I(a,M\beta)<\infty. \tag{34}\] It follows that \(Q_t\exp(-\int_0^t k_v\,\mathop{}\!\mathrm{d}v)\) is a nonnegative local supermartingale on every nonsingular interval, including its downward activation and completion jumps.

Integrating the collision singularities

We record precisely the moment consequence used in the next sections. For \(l>0\) let \[A_l(\mathop{}\!\mathrm{d}x)=G_{\tau_l(x)}(x) \mathbf 1_{\{\tau_l(x)<\infty\}}\,\mathop{}\!\mathrm{d}A(x).\]

Deterministic convex extraction

The compact moment bounds permit one simultaneous limit for all rational times and a determining family of spatial tests. Keeping the same convex coefficients is what later makes the kernels and conformal images properties of one measure rather than separately chosen versions.

For an integer \(j\ge1\) define \[ M_t^{(j)}(\mathop{}\!\mathrm{d}x) =G_{\tau_j(x)}(x) \mathbf 1_{\{\tau_j(x)\le t,\ \tau_j(x)<\infty\}}\,\mathop{}\!\mathrm{d}A(x), \qquad 0\le t\le\infty. \tag{41}\] The density is zero on failure to reach the level. For a compact \(C\Subset\mathbb H\), fix \(c_C>0\) with \(c_C\le\inf_C R_0\). Given distinct \(x_1,\ldots,x_m\in C\), set \[b_i=\min\bigl(c_C,\min_{k\ne i}|x_i-x_k|\bigr),\] where the second minimum is omitted when \(m=1\). The terminal approximant is the level weight \(A_j\) of Corollary 7. The collision-kernel estimate in its proof gives, for every bounded nonnegative \(f\) supported in \(C\), \[ \sup_j\mathbb E\bigl[M_\infty^{(j)}(f)^m\bigr] \le C_{m,a}\int_{C^m} \prod_{i=1}^m f(x_i)b_i^{-\alpha} \,\mathop{}\!\mathrm{d}A(x_1)\cdots\mathop{}\!\mathrm{d}A(x_m)<\infty. \tag{42}\] The last assertion is Lemma 6. All constants are independent of \(j\).

We obtain one sequence of convex combinations as follows. Choose a countable family of continuous compactly supported tests, uniformly dense among tests supported in each member of a compact exhaustion. Include nonnegative cutoff functions equal to one on that exhaustion, and rational linear combinations of the chosen tests. Enumerate the pairs of these tests with times in \(\mathbb Q_+\cup\{\infty\}\) as \((f_r,t_r)\), \(r\ge1\), and set \(X_r^{(j)}=M_{t_r}^{(j)}(f_r)\). Each coordinate is bounded in \(L^2\) by (42). A diagonal subsequence \(j_l\) has a weak limit \(X_r\) in every coordinate.

For each \(k\), consider the first \(k\) coordinates as a vector in \((L^2)^k\). Its weak limit \((X_1,\ldots,X_k)\) lies in the norm closure of the convex hull of every tail. Hence there are deterministic coefficients \(c_{kl}\ge0\), supported on finitely many \(l\ge k\), with \(\sum_lc_{kl}=1\), such that \[\sum_{r=1}^k\left\| \sum_{l\ge k}c_{kl}X_r^{(j_l)}-X_r \right\|_2^2\le4^{-k}.\] Define \[\overline M_t^{(k)}=\sum_{l\ge k}c_{kl}M_t^{(j_l)}.\] For every listed pair, the summable squared errors give both \(L^2\) and almost-sure convergence; intersecting the resulting events over \(r\) gives simultaneous convergence on the whole countable list. The same coefficient row is used for every time and spatial test.

On this event, the cutoff tests bound all local masses. Density therefore extends convergence to every continuous compactly supported test. The limiting positive functionals are Radon measures, with almost-sure vague convergence at all listed times. Positivity and ordering in time pass to the limit. The limits at finite rational times are adapted, since the corresponding \(L^2(\mathcal F_t)\) subspaces are closed. Convexity and Fatou’s Lemma preserve (42) for the terminal measure and continuous nonnegative tests. Approximation then gives the compact moment assertions.

We also obtain nonatomicity, which will be needed to pass from rational times to all times. For two points in a fixed larger compact the kernel in (42) is bounded by \(C(1+|x-y|^{-2\alpha})\). Cover a given smaller compact by mesh squares of side \(\delta\), and majorize each square’s indicator by a continuous cutoff in a slightly enlarged square. The enlarged squares have bounded overlap. The expected sum of their squared masses is bounded by \[C\iint_{|x-y|\le C\delta} (1+|x-y|^{-2\alpha})\,\mathop{}\!\mathrm{d}A(x)\mathop{}\!\mathrm{d}A(y),\] where both variables lie in the larger compact. This tends to zero because \(2\alpha<2\). Every atom in the smaller compact contributes the square of its mass to the corresponding sum at every mesh size. Fatou’s Lemma, followed by a countable compact exhaustion, proves that \(\mu_\infty\) has no atoms.

Support, continuity, and stopping times

The finite-level conditional weight identity of Lemma 2 gives, at a finite rational time \(t\), \[ \mathbb E\bigl[M_\infty^{(j)}(\mathop{}\!\mathrm{d}x)-M_t^{(j)}(\mathop{}\!\mathrm{d}x) \mid\mathcal F_t\bigr] =G_t(x)\mathbf 1_{\{x\in H_t,\ R_t(x)>R_0(x)e^{-2aj}\}}\,\mathop{}\!\mathrm{d}A(x). \tag{43}\] The strict inequality excludes a level already attained at \(t\). At every surviving point the indicator tends to one. Since \(\mathbb EG_t(x)\le G_0(x)\), dominated convergence on compact tests applies to the right side. Local \(L^2\) bounds give uniform integrability on the left. Passing through the tail convex combinations proves (39) at rational times. The mean-one finite-level weights also give (40).

These identities initially hold for countably many deterministic tests. They identify the conditional measure kernel, and hence, by the Monotone Class Theorem, hold against every nonnegative \(\mathcal F_t\otimes\mathcal B(\mathbb H)\)-measurable test. In particular, testing the complement of the known surviving domain shows that \[ \mu_\infty-\mu_t\text{ is carried by }H_t \quad\text{at every finite rational }t, \tag{44}\] on one event of probability one.

At the same times, \(\mu_t\) is carried by the past trace. Indeed, on a fixed compact, a point receiving mass in \(M_t^{(j)}\) has, at its level time, distance at most \(R_0(x)e^{-2aj}\) from the current domain boundary. For large \(j\) that nearby boundary cannot be the real line, so it belongs to \(\gamma([0,t])\). Thus the approximants, and their tail convex combinations, are carried locally by shrinking neighborhoods of this closed trace segment. Vague convergence gives the claimed support. For rational \(v<t\), the increment is dominated by \(\mu_t\) and by \(\mu_\infty-\mu_v\), so it is carried by \(\gamma([v,t])\cap\mathbb H\).

Take increasing rational left limits and decreasing rational right limits of the measure process. At any finite time \(w\), their difference is dominated by increments over every rational interval containing \(w\). Continuity of the trace forces its support to lie in the singleton \(\{\gamma(w)\}\cap\mathbb H\). The difference vanishes by nonatomicity. At zero use zero as the left limit; the same argument gives zero, since \(\gamma(0)\notin\mathbb H\). This argument is pathwise on the one event already constructed and therefore works simultaneously for every \(w\). It defines an increasing process on all times, locally continuous in total variation: on a compact the variation of a positive increment is just its mass, and the possible one-sided jump measures have been shown to vanish. Adaptedness follows from the rational left limits. The support assertion for arbitrary intervals follows by enclosing them in rational intervals and taking limits.

The terminal value is exhausted by finite times. By Proposition 4, \(\mathbb EG_t(x)\to0\) as \(t\to\infty\). Apply (39) at integer times and use \(\mathbb EG_t\le G_0\) on compact tests. The nonnegative measure \(\mu_\infty-\lim_{t\to\infty}\mu_t\) has zero mean, and hence is zero. This also gives local total-variation convergence at infinity.

For completeness, let \(S\) be any stopping time. On the event of rational convergence, monotonicity squeezes \(\overline M_S^{(k)}(f)\) between its values at rational times approaching each realized finite \(S\). Continuity gives convergence to \(\mu_S(f)\) for every nonnegative continuous compact test. On \(\{S=\infty\}\) use terminal convergence of the extracted sequence. Uniform integrability again follows from the terminal \(L^2\) bound. The finite-level identity (43) holds with \(t\) replaced by \(S\), and with zero right side on \(\{S=\infty\}\). The bound \(\mathbb E[G_S(x);S<\infty]\le G_0(x)\) from Proposition 4 permits the same dominated convergence. This proves (39) at \(S\), including its conditional-kernel interpretation, and completes Proposition 8.

Versions, conformal maps, and restart

We have constructed the continuous mass and its conditional kernel. To use them at centered stops and in random future domains, we next fix a measurable path functional and establish equality of the actual restarted measures.

We fix one nonanticipating measurable version of this construction. On the canonical continuous-path space each \(M^{(j)}\), tested on an interior compact, is a raw adapted càdlàg process: its terminal density has a deterministic finite bound there, and one-sided continuity follows from dominated convergence of the hitting-time indicators. The same holds for every deterministic convex combination. At each finite time, take the vague limit of \(\overline M_t^{(k)}\) where it exists, and use the zero measure otherwise. The space of locally finite positive Radon measures is Polish in the vague topology; its convergence set and limit map are therefore Borel. This defines a raw progressively measurable measure-valued functional. At infinity use the same limit rule for \(\overline M_\infty^{(k)}\). Rational squeezing as above shows that it agrees, simultaneously at every time under the ordinary Brownian law, with the continuous process just constructed.

All evaluations under a centered law use this raw functional of the stopped path. At each finite inner level the density in Lemma 2 transfers the required statements about that stopped piece. Consistent laws of the stopped pieces, and increasing limits of their coordinates, suffice thereafter. We do not extend the centered law to the original Brownian space completed by infinite-future null sets. This distinction also ensures parameter measurability: Loewner solutions, level times and conformal maps are measurable in the driver and marked point, and progressive evaluation of the fixed mass functional at those times is measurable.

For a fixed conformal identification \(J:\mathbb H\to D\), define \[ \mu_t^D=J_*\bigl(|J'|^d\mu_t\bigr). \tag{46}\] This rule holds exactly for the approximants: conformal radius contributes \(|J'|^{-\alpha}\) and area contributes \(|J'|^2\), so their product is \(|J'|^d\). Inner-clock ratios are unchanged. A compact continuous test in \(D\) pulls back to such a test in \(\mathbb H\). Thus the same deterministic convex combinations converge in every fixed image domain, on the event giving convergence for all compact continuous tests. The kernel, local continuity, nonatomicity and support statements hold there with the corresponding \(G_t^D\). More precisely, its increments are carried by \(J(\gamma([v,t])\cap\mathbb H)\). The map is applied only to interior trace points; an arbitrary \(J\) need not extend continuously to the boundary. The construction is jointly measurable in a measurable family of maps \(J\); it requires no new extraction for different maps.

The kernel also proves uniqueness. If \(A_t\) and \(\widetilde A_t\) are two continuous increasing adapted test-mass processes starting at zero, with integrable terminal values and the same remaining conditional potential, then \[A_t-\widetilde A_t =\mathbb E[A_\infty-\widetilde A_\infty\mid\mathcal F_t].\] Their difference is a continuous martingale of finite variation, hence vanishes. Applying this to a countable determining set of tests proves uniqueness of the measure process. In particular it removes dependence on the convex extraction. It also gives Brownian scaling: if \(B^{(c)}_t=c^{-1}B_{c^2t}\), then, for each \(c>0\), \[ \mu_t(B^{(c)})(A)=c^{-d}\mu_{c^2t}(B)(cA) \quad\text{almost surely, for all }t\text{ and Borel }A\subset\mathbb H. \tag{47}\] Indeed the scaled process has the same kernel, continuity and terminal integrability as the canonical process for \(B^{(c)}\).

For natural parametrization, Lawler–Rezaei (Lawler and Rezaei 2012, Proposition 3.13) prove conformal restart at an almost surely finite stopping time, simultaneously over later time intervals. We prove the following measure-valued identity for the mass constructed here directly from its conditional kernel.

To obtain the centered restart rule, first work under the ordinary law. For a stopping time \(S\), work on \(\{S<\infty,\ z\in H_S\}\) and write \(g_S(z)-B_S=\lambda e^{i\theta}\), where \(\lambda>0\). Conditionally on \(\mathcal F_S\), the rescaled fresh driver \(\lambda^{-1}(B_{S+\lambda^2u}-B_S)\) is standard Brownian motion. We may apply (47) at this random scale: after conditioning, \(\lambda\) is fixed, the fresh driver is independent of \(\mathcal F_S\), and the scaling equality holds almost surely at each fixed scale. Joint measurability of the mass functional permits this conditioning. Together with Proposition 10, scaling identifies the actual future mass in these coordinates.

Now apply the finite change of law through the next radius target of \(z\). The ratio of Green weights is conformally invariant, since the derivative at the marked point cancels between the two weights. Lemma 2 therefore identifies, conditionally on the stopped past under the centered law, the future mass pulled back to a disk centered at \(z\) with the canonical disk law of starting angle \(\theta\). This assertion uses only data through the finite test endpoint.

The centered disk model

For \(0<\theta<\pi\), use the conformal identification \(J_\theta:\mathbb H\to\mathbb D\) below. It takes \(e^{i\theta}\) to the marked center \(0\). Denote the ordinary law in these coordinates by \(\mathbb P_\theta\), and its centered finite-level laws at \(0\) by \(\mathbb P_\theta^*\). Average the latter over the invariant angle law \(\pi_a\), and write \(\mathbb P_{\mathrm{mix}}^*\) for the resulting consistent stopped laws. Define \[ \begin{gathered} J_\theta(\zeta)=\frac{\zeta-e^{i\theta}}{\zeta-e^{-i\theta}}, \qquad K=\{x\in\mathbb D:|x|\le\tfrac14\}, \qquad \mathbb P_{\mathrm{mix}}^*=\int_0^\pi\mathbb P_\theta^*\,\pi_a(\mathop{}\!\mathrm{d}\theta), \\ L_\ell=\mu^{\mathbb D}_{\tau_\ell(0)}(K), \qquad L=\lim_{\ell\to\infty}L_\ell. \end{gathered} \tag{50}\] The disk mass is the weighted pushforward of the single canonical half-plane functional in Section 4. Each \(L_\ell\) is a raw finite-stopped-path quantity; its increasing limit is defined on the consistent collection of such quantities. In particular (50) does not require a Brownian continuation after the centered terminal time.

A Wiener-integral formula

We use the discrete potential compensator appearing in Lawler–Zhou (Lawler and Zhou 2013, Theorem 1 and its proof). We prove its convergence at a random radius level and then combine it with the finite-level Green weight to specify the centered moments.

Let \(W\) be standard Wiener measure on \(C_0([0,\infty),\mathbb R)\). For a fixed \(\theta\) and driver \(b\), solve the ordinary Loewner equation with numerator \(a\), using \(J_\theta\) for the disk identification, and form \[\Psi_t(\theta,b)=\int_K G_t(x;\theta,b)\,\mathop{}\!\mathrm{d}A(x), \qquad \tau_\ell=\tau_\ell(0), \qquad P_\ell(\theta,b) =\mathbf 1_{\{\tau_\ell<\infty\}}e^{2a\alpha\ell} \frac{\sin^p\theta_{\tau_\ell}(0)}{\sin^p\theta}.\] The last expression is zero on failure. For paths \(b,c\), let \(b\oplus_t c\) agree with \(b\) through time \(t\), and equal \(b(t)+c(\,\cdot-t)\) thereafter. Put \[\delta_h=2^{-h},\qquad N_h=h2^h,\qquad t_i=i\delta_h,\] and define the potential differences and their stopped cumulative sum by \[\begin{align*} q_{h,i}(\theta,b) &=\Psi_{t_i}(\theta,b) -\int\Psi_{t_{i+1}}(\theta,b\oplus_{t_i}c)\,\mathop{}\!\mathrm{d}W(c), \tag{54}\\ D_h(\theta,b;\ell) &=\sum_{i=0}^{N_h-1}\mathbf 1_{\{t_{i+1}\le\tau_\ell\}} q_{h,i}(\theta,b). \tag{55}\end{align*}\] Set undefined differences to zero. For every fixed \(\theta\), these exceptions have Wiener measure zero by the compact first moment and the conditional kernel. All integrands are measurable in \(\theta\) and the paths. Write \([x]_0^u=\min\{\max\{x,0\},u\}\).

The next estimate explains why a potential formula recovers mass at a random endpoint. For fixed \(\theta\), write \(A_t=\mu_t^{\mathbb D}(K)\) and interpret \(A_{\tau_\ell}\) as \(A_\infty\) when \(\tau_\ell=\infty\).

A prescription using only normal coordinates and ODE limits

The Wiener integrals in (57) are deterministic integrals. The following construction specifies their integrands without using a sample of the trace in Theorem 1. They can be realized as countable product standard-normal integrals: choose each unit increment of \(b\) as an independent standard normal, and recursively set the midpoint of a dyadic interval of length \(2^{-k}\) equal to the average of its endpoint values plus \(2^{-k/2-1}\) times a new independent standard normal. The polygonal interpolants converge uniformly on compacts almost surely. Indeed Gaussian tails and Borel–Cantelli bound the largest level-\(k\) normal on finitely many unit intervals by \(O(\sqrt{k+1})\), and \(\sum_k2^{-k/2}\sqrt{k+1}<\infty\). The limit has Gaussian covariance \(\min\{s,t\}\) and hence Wiener law. Use the zero path on the null exceptional set.

The Loewner quantities in the integrands do not require constructing the trace. For \(\zeta=J_\theta^{-1}(x)\) and an integer \(j\ge1\), use the truncated vector field \[F_j(t,z)=\frac{a\,\overline{z-b(t)}} {\max\{|z-b(t)|^2,j^{-4}\}}.\] Starting at \(\zeta\), perform the Euler iteration \(g_{k+1}=g_k+2^{-N}F_j(k2^{-N},g_k)\) and interpolate linearly. For fixed \(j\), its compact-uniform limit \(g^{(j)}\) exists as \(N\to\infty\): the vector field is bounded, continuous in time, and globally Lipschitz in its state variable. A given finite time \(t\) is surviving exactly when some \(j\) satisfies \[\min_{0\le u\le t}\operatorname{Im}g_u^{(j)}>j^{-1}.\] For such a witness, \(|g^{(j)}-b|^2>j^{-2}\ge j^{-4}\), so the cutoff is inactive and uniqueness gives the genuine Loewner solution on \([0,t]\). Conversely, a surviving solution on this compact interval has positive minimum imaginary part, and a sufficiently large \(j\) recovers it.

On survival use this solution to compute \[\begin{align*} s_x(t)&=\int_0^t\frac{(\operatorname{Im}g_u)^2}{|g_u-b(u)|^4}\,\mathop{}\!\mathrm{d}u,\\ R_t(x)&=2\operatorname{Im}\zeta\,|J_\theta'(\zeta)|e^{-2as_x(t)},\\ G_t(x)&=R_t(x)^{-\alpha}\sin^p\arg(g_t-b(t)), \end{align*}\] and put \(G_t(x)=0\) off survival. At the marked center the level time is \[\tau_\ell= \inf\{q\in\mathbb Q_{>0}:q\text{ is surviving and }s_0(q)>\ell\}, \qquad \inf\varnothing=\infty.\] Continuity and strict increase of the clock on its open lifetime identify this infimum with attainment of level \(\ell\) while alive. These rational tests, deterministic limits and integrals specify every integrand in (57). This is an infinite-limit prescription; no finite computational error bound is asserted. The coefficients retain neither an unspecified normalization of trace mass nor a choice of convex extraction.

The finite lists and their radii

For integers \(n\ge1\) and \(k\in\mathbb Z\), put \[ \ell_n=1+\sum_{j=1}^\infty \mathbf 1_{\{m_n(j)\le 3m_n/4\}}, \qquad A_k=2^k, \qquad T_n=\frac{4m_{n+1}}{m_n}. \tag{58}\] The sum is finite, and \(m_n(\ell_n)>3m_n/4\), since \(m_n(j)\uparrow m_n\in(0,\infty)\). Form a list, in increasing order of \(k\), containing \(A_k\) with multiplicity \[ N_{nk}=\mathbf 1_{\{2^k\le T_n\}} \left\lfloor\frac{4^n2^{kn}}{m_n}\right\rfloor. \tag{59}\] Append the number \(n\) once, as the final cleanup entry, and denote the resulting entries by \(b_{ni}\), \(1\le i\le I_n\). This list is finite. Indeed, \(N_{nk}>0\) implies \[\frac{m_n^{1/n}}4\le A_k\le T_n,\] and Jensen’s Inequality gives \(m_n^{1/n}\ge m_1>0\). Thus \(c_0:=\min\{1,m_1/4\}\) is a positive lower bound for every entry \(b_{ni}\).

Assign the radii \[ r_{ni}=\exp\{-n^2-2a(i-1)(\ell_n+n)\}, \qquad 1\le i\le I_n, \tag{60}\] and define \[ \boxed{\quad h_\kappa(r)= \inf_{\substack{n\ge1\\1\le i\le I_n}} b_{ni}r_{ni}^{\,d}\max\{1,(r/r_{ni})^2\} \quad(r>0),\qquad h_\kappa(0)=0. \quad} \tag{61}\] The infimum includes every entry of every batch.

The quadratic envelope has a planar covering antecedent in Peres–Solomyak (Peres and Solomyak 2005, Lemma 1.2 and Section 5). Indeed, for \(0<s\le e^{-1}\) put \(\phi(s)=\inf_{r_{ni}\ge s}b_{ni}r_{ni}^{d}\). Exchanging infima gives \[h_\kappa(s)=\inf_{0<r\le s}(s/r)^2\phi(r),\] which is their quadratic regularization of this step profile. The batch data, nondegeneracy of the envelope, and control after finite exclusions are established here; in particular, positivity is proved directly rather than inferred for an arbitrary input profile.

Admissibility and restoration of early entries

The envelope must remain positive at every positive diameter and tend to zero at the origin. Its behavior after a finite set of entries is excluded will also be needed in the lower-cover proof.

The opposing probability estimates

The multiplicities supply the two bounds advertised at the start of this section. These are estimates under the one mixed disk law; the conditional tests along the curve are constructed in Section 8.

A spatial comparison at a stopped restart

Fix the numerical constants \[ B_*=100,\qquad C_*=200,\qquad c_*=\frac{192}{5},\qquad C_*'=\frac{8000}{27}. \tag{66}\] For \(z\in\mathbb H\) and \(100r<R_0(z)=2\operatorname{Im}z\), let \(S\) be the first time that \(R_S(z)=100r\) while \(z\) remains alive. On \(\{S<\infty\}\), choose a disk map \(F:\mathbb D\longrightarrow H_S\) with \(F(0)=z\), normalized so that the restarted half-plane coordinate at \(z\), after positive rescaling, is \(e^{i\theta_S(z)}\). Thus the disk has exactly the markings of Equation (50), and \(|F'(0)|=100r\). For \(K=\{|w|\le1/4\}\), the growth and distortion estimates give \[\begin{align*} \overline B(z,16r)&\subset F(K) \subset\overline B\left(z,\frac{400}{9}r\right) \subset\overline B(z,C_*r), \tag{67}\\ c_*r&\le |F'(w)|\le C_*'r\qquad(w\in K). \tag{68}\end{align*}\] Indeed the inner and outer growth bounds are \(100r(1/4)/(1\pm1/4)^2\), and the derivative bounds are \(100r(1-1/4)/(1+1/4)^3\) and \(100r(1+1/4)/(1-1/4)^3\). In particular, \(F(K)\) contains every closed grid square of side \(r\) containing \(z\).

Since \(F(K)\) is compactly contained in \(H_S\), its accumulated mass at time \(S\) is zero. Proposition 10 identifies the actual subsequent measure there, simultaneously at all later times, with the pushforward of the disk measure weighted by \(|F'|^d\). Consequently, if \(T\) is the time of an additional inner-clock increment \(0<\ell<\infty\) at \(z\), then, on \(\{S<T<\infty\}\), \[ (c_*r)^d L_\ell \ \le\ (\mu_T-\mu_S)(F(K)) \ \le\ (C_*'r)^d L_\ell \tag{69}\] in these restarted coordinates. The event that this test endpoint is reached has probability one under the corresponding finite centered law. Conditional on the stopped past under \(\mathbb P_z^*\), the variable on the right is the disk-model variable with initial angle \(\theta_S(z)\). This is an equality of actual measures before applying distortion, so it also permits random Borel sets and random derivative weights. The clock increment is exactly \(\ell\): the derivative at \(z\) cancels from ratios of conformal radii.

All uses of Equation (69) under a centered law are first made on a finite stopped piece, using the raw mass functional of Proposition 8. For example, if \(q<T^*=\lim_{\ell\to\infty}\tau_\ell(z)\), some finite stopped piece extends beyond \(q\). Monotonicity therefore bounds the mass accumulated in \(F(K)\) before \(q\) by \((C_*'r)^dL\), with \(L\) the increasing limit in the restarted disk model. No mass after \(T^*\) is used.

Past mass and first visits

The relation between natural-length-biased chordal SLE and integrated Green-weighted two-sided radial laws was proved by Field (Field 2016, Theorem 1) for \(0<\kappa\le4\) in bounded domains with analytic boundary, and by Zhan (Zhan 2019a, Theorem 4.1) for \(0<\kappa<8\). Zhan’s proof connects mass strictly after an observation time to centered stopped laws through the conditional remaining Green potential (Zhan 2019a, Proposition 2.2, equation (2.1), and equations (4.3)–(4.5)). We use this mechanism to derive the required past-mass identity directly for our canonical mass, from its conditional measure kernel and finite stopped changes of law. Retaining the strict time inequality keeps each centered-law mass evaluation before its terminal time.

Let \(Q\Subset\mathbb H\) be a deterministic closed square and let \(H>0\). Write \(M_q=\mu_q(Q)\), and let \(v(z)\) denote the first visit time of \(z\). We shall use the following identity: \[\begin{align*} &\mathbb E\int_Q \mathbf 1_{\{\exists q\in\mathbb Q_{\ge0}: \ q<v(z),\ M_q>H\}}\,\mu_\infty(\mathop{}\!\mathrm{d}z) \\ &\hspace{12mm}= \int_Q G_0(z)\, \mathbb P_z^*\{\exists q\in\mathbb Q_{\ge0}: \ q<T^*,\ M_q>H\}\,\mathop{}\!\mathrm{d}A(z). \tag{70}\end{align*}\] The strict inequality \(q<T^*\) is part of the statement. Here is a derivation that also specifies the measurable kernel on its right-hand side.

First take a finite set of rational times \(q_1<\cdots<q_N\), and set \[A_j=\{M_{q_j}>H,\ M_{q_k}\le H\text{ for }k<j\}.\] Each \(A_j\) belongs to \(\mathcal F_{q_j}\) and does not depend on the integration point \(z\). If a successful test in this finite set precedes \(v(z)\), its first successful test also precedes \(v(z)\). The first-visit identity in Corollary 9 thus identifies the corresponding random mass with \[\sum_{j=1}^N \mathbf 1_{A_j}(\mu_\infty-\mu_{q_j})(Q).\] Apply Equation (39) with the event \(A_j\), followed by the past-test identity in Proposition 4. The expectation becomes \[\int_Q G_0(z) \sum_{j=1}^N\mathbb P_z^*(A_j,\ q_j<T^*)\,\mathop{}\!\mathrm{d}A(z).\] These events are disjoint. If a later successful test occurs before \(T^*\), then the first successful test does too.

For completeness, for a fixed \(A\in\mathcal F_q\) the term just used has the measurable version \[ \mathbb P_z^*(A,q<T^*)= \lim_{\ell\to\infty} \frac{1}{G_0(z)} \mathbb E\!\left[ \mathbf 1_A G_q(z) \mathbf 1_{\{z\in H_q,\ s_z(q)<\ell\}} \right]. \tag{71}\] At a finite level the expectation also equals \(\mathbb E[\mathbf 1_A\mathbf 1_{\{q<\tau_\ell(z)<\infty\}} G_{\tau_\ell(z)}(z)]\). This is a stopped-past test of the finite tilted law. The raw mass functional is fixed once, and the Loewner quantities are jointly measurable in \(z\) and the driving path. Thus Equation (71) is a measurable function of \(z\), and Tonelli’s Theorem applies without a common exceptional set for uncountably many marked points. Finally, exhaust \(\mathbb Q_{\ge0}\) by increasing finite sets, sorting each set before making the first-success selection. The existence events increase even though their finite partitions change. Monotone convergence proves Equation (70).