= Cauchy coupling proves continuity of the half-plane Poisson extension
{c}
{title2=$\phi(x+iy)=\mathbb E f(x+yC),\quad C\sim\operatorname{Cauchy}(0,1)$}
For bounded continuous boundary data $f$, the half-plane <Poisson kernel> is the law of $x+yC$. On one common <probability space>, $x+yC\to x_0$ almost surely as $(x,y)\to(x_0,0)$ with $y>0$. Boundedness of $f$ and <dominated convergence theorem> imply $\phi(x+iy)\to f(x_0)$. This proves <continuity> at every finite boundary point even for tangential approach. No moment of the <Cauchy distribution> is needed.
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