= Solution
Interpret the printed $A_0=0$ as the empty initial hull. Write $a(t)=\operatorname{hcap}(\widetilde A_t)$ and $F_t=\psi_t$. <Conformal maps> preserve inclusion and simple connectedness of the complementary domains, so the image sets form an increasing hull family. Boundedness of $\psi$ on bounded sets ensures that these image hulls are bounded. The <Loewner local growth property> is preserved under conformal transport: after mapping out the hull at time $t$, the small new hull is transported by $F_t$ near its single boundary growth point. The <Schwarz reflection principle> extends $F_t$ analytically across that point, with real positive derivative. The image diameters therefore tend to zero as the original ones do. These statements use the usual hull closures and <prime ends>; the initial image hull is empty.
For completeness, the infinitesimal capacity rule underlying this argument is that a shrinking hull $J$ attached near $u$, transported by a map $F$ analytic there, has
$$
\operatorname{hcap}(F(J))=F'(u)^2\operatorname{hcap}(J)+o(\operatorname{hcap}(J)).
$$
One obtains this by rescaling at $u$: the transported map tends to its linear part, and <half-plane capacity> scales by the square of the dilation. Uniform analytic distortion near the growth point controls the error. Applying it to the mapped-out increments gives local absolute continuity of $a$ and $a'(t)=2F_t'(U_t)^2$.
The coefficient can also be read directly from the <Chordal Loewner equation>. The image driver is $\widetilde U_t=F_t(U_t)$. Differentiate $F_t=\widetilde g_t\circ\psi\circ g_t^{-1}$, at a fixed point $w$, to obtain
$$
\partial_tF_t(w)=\frac{a'(t)}{F_t(w)-F_t(U_t)}-\frac{2F_t'(w)}{w-U_t}.
$$
The left side is regular at $w=U_t$. On the right, the coefficient of $(w-U_t)^{-1}$ is $a'(t)/F_t'(U_t)-2F_t'(U_t)$, so it must vanish. This proves the <conformal change of half-plane capacity> rule and its integrated version:
$$
\boxed{\operatorname{hcap}(\widetilde A_t)=\int_0^t2\bigl(\psi_s'(U_s)\bigr)^2\,ds,\qquad\widetilde A_0=\varnothing.}
$$
If the derivative at the initial boundary point is singular, the formula is interpreted by integrating from a positive time and taking the lower limit to zero; finite image capacity and kernel continuity give this limit. The duplicated incomplete normalization sentence in the TeX is absent from the PDF.
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