= Solution
The construction is well defined. Since $D$ is contained in a ball, its <Brownian exit time> is finite <almost surely>: at successive integer times there is a fixed positive probability that the next independent unit-time <Brownian motion> increment has length greater than the ball's diameter, forcing an exit. The survival probability is therefore bounded by a geometric sequence. Path continuity gives $B_{\tau_D}\in\partial D$. Thus the bounded <Dirichlet boundary data> give a bounded Borel function $u$ on $D$.
For interior harmonicity, fix a ball with closure in $D$, centred at $x$, and let $\sigma$ be its <Brownian exit time>. The <Strong Markov property>, followed by the <tower property of conditional expectation>, gives
$$
u(x)=\mathbb E_x[u(B_\sigma)].
$$
The <orthogonal invariance of Brownian motion> makes $B_\sigma$ uniform on the boundary sphere. Hence $u$ has the spherical <mean value property for harmonic functions> for every such ball. A locally bounded Borel function with this property is smooth and harmonic: integrating the spherical averages against any smooth radial <mollifier> gives $u=u*\rho$ locally, which first proves smoothness; the <mean value property for harmonic functions> then implies $\Delta u=0$.
For the boundary limit at $\xi$, choose the harmonic <barrier for the Dirichlet problem> $h_\xi$ from (i). The <Itô formula>, localized inside $D$, makes $h_\xi(B_{t\wedge\tau_D})$ a bounded <martingale>. Compact localization and the <bounded convergence theorem>, first to the exit and then as $t\to\infty$, yield
$$
\mathbb E_x h_\xi(B_{\tau_D})=h_\xi(x).
$$
For $\varepsilon>0$, if $\partial D\setminus B(\xi,\varepsilon)$ is nonempty, <compactness> and barrier positivity give
$$
c_\varepsilon=\min_{\eta\in\partial D,\ |\eta-\xi|\geq\varepsilon}h_\xi(\eta)>0,\qquad
\mathbb P_x(|B_{\tau_D}-\xi|\geq\varepsilon)\leq\frac{h_\xi(x)}{c_\varepsilon}\longrightarrow0\quad(x\to\xi).
$$
If that boundary subset is empty, the probability is already zero. Therefore
$$
|u(x)-f(\xi)|\leq\sup_{\eta\in\partial D,\ |\eta-\xi|<\varepsilon}|f(\eta)-f(\xi)|+2\|f\|_\infty\mathbb P_x(|B_{\tau_D}-\xi|\geq\varepsilon).
$$
First let $x\to\xi$ and then $\varepsilon\downarrow0$. The <continuity> of the <Dirichlet boundary data> proves $u(x)\to f(\xi)$.
Finally, the difference of two continuous solutions is a <harmonic function> vanishing on the boundary. The <maximum principle for harmonic functions> on the bounded domain gives that it is zero. Thus \b[the <Kakutani solution of the Dirichlet problem> exists, attains every prescribed boundary value, and is unique].
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