Exercise 11.14

Question

Prove that a complex function $f$ is measurable if and only if $f^{-1}(V)$ is
    measurable for every open set $V$ in the plane.

Amit Awasthi · Accepted Answer

\begin{proof}
      Let $f = u + iv$. Then $f$ is measurable if and only if both $u$ and $v$
      are measurable.
      \par Now the $\Leftarrow$ direction is trivial since letting $V =
      (a,\infty) + i(b,\infty)$, we see that $f^{-1}(V) = u^{-1}(A,\infty)
      + iv^{-1}(b,\infty)$ is measurable.
      \par Conversely, if $V$ is an arbitrary open set in the plane, we can tile
      it by a countable number of rectangles, and a similar argument as above
      works.
    \end{proof}

Exercise 11.14

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Exercise 11.14

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