OAI.SLEExactGauge.sourceLowerMain_proved

The theorem states that, for every probability space (Ω, P) and every κ with 0<κ<8, if γ is an ordinary chordal SLE_κ family of curves on Ω, then almost surely the explicit Hausdorff gauge measure of every positive-time segment of the curve is positive. Here γ being an ordinary chordal SLE_κ means that each γ(ω,t) is measurable in ω and that there is a standard real Brownian motion B on (Ω,P) such that, for almost every ω, the curve t↦γ(ω,t)∈ℂ with driving function U(t)=√κ·B_t(ω) is a capacity-two trace: γ is continuous, starts at 0, stays in the closed upper half-plane, and there are conformal-type maps G_t from the unbounded component of the upper half-plane minus γ[0,t] to the upper half-plane and inverse maps F_t back, holomorphic, mutually inverse, with G_0 the identity, satisfying the Loewner equation ∂_t G_t(z)=2/(G_t(z)−U(t)) (one-sided at t=0), the normalization z(G_t(z)−z)→2t as z→∞, and F_t(U(t)+iy)→γ(t) as y→0⁺. The gauge h:ℝ→ℝ is any function that is continuous, nondecreasing on [0,∞), zero at 0 and positive for r>0, and that coincides, for all sufficiently small r>0, with r^{1+κ/8}·(log log(1/r))^{(1−κ/8)/2}. The gauge measure is the Hausdorff-type metric measure on ℂ built from r↦h(r) (with the ENNReal-valued radius cast through its real part). The conclusion is that, for almost every ω, simultaneously for all times 0<s<t, this measure of the image set γ(ω)([s,t]) is strictly positive.