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How to integrate an uncountable collection of points in the complex plane
I am trying to prove that if $f$ is continuous on the closed disk $overline{mathbb{D}}$ and if the set of all the complex numbers $z$ for which $f(z)=0$ has Lebesgue measure $0$, then $f(z)=0$ for every $zin overline{mathbb{D}}$. I haven’t been able to think of a way to do this.
A:
Let $z$ be an element of the closed disk. If you consider $mathbb{D}$ as a subspace of the complex plane, then it has the usual topology. Then as $f$ is continuous on $mathbb{D}$, there exists $r>0$ such that $f(B(z,r))subset f(mathbb{D})=0$, where $B(z,r)$ is the open ball with center $z$ and radius $r$.
The set $left{zinmathbb{D}:f(z)=0
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