Simple proof of a theorem on removable singularities of analytic functions satisfying a Lipschitz condition

Let $E$ be a compact subset of the complex plane $\mathbb C$, having positive planar Lebesgue measure. Then there exists a nonconstant function $f$, analytic in the domain $\mathbb C\setminus E$, satisfying the Lipschitz condition
\begin{equation} |f(z_1)-f(z_2)|\le\operatorname{const}|z_1-z_2|,\qquad z_j\in\mathbb C\setminus E,\quad j=1,2. \end{equation}
In this note there is given a simple proof of the theorem of N. X. Uy, formulated above. It is also proved that each bounded measurable function $\alpha$, defined on the set $E$, can be revised on a set of small ebesgue measure so that for the function $\varphi$ obtained the Cauchy integral
$$ f(z)=\iint_E\frac{\varphi(t)}{t-z}\,dm_2(t) $$
satisfies condition (1).