 Solution

Write $r_t=\sqrt{2t}$. The hulls $A_t=r_t(\mathbb{H}\cap \mathbb{D})$ are nested, so property (i) holds. Scaling the given map givesso $\mathrm{hcap}(A_t)=2t$ and property (ii) holds.

Property (iii) fails. For $s>0$, the image under $g_s$ of the outer semicircle of $A_t\setminus A_s$ isAs $t\downarrow s$, this converges to $2r_s\cos \theta$, which fills the real interval $[-2r_s,2r_s]$. Hence the diameter of the mapped new increment tends to $4r_s$, rather than zero. Therefore (i) and (ii) hold, while (iii) does not.