Based on the answer here, the following result is clear:
``Given a complex analytic function $f(x)$ with $x$ is in an open connected set $D \subseteq \mathbb{C}^n$, the zero set $$F = \{x \in D | f(x) = 0\}$$ has $2n$-dimensional Lebesgue measure zero"
Now consider a slightly different situation:
A complex analytic function $g(z)$ with $z \in D \times V$, where $D$ is an open connected subset of $\mathbb{C}^n$ and $V$ is an open connected subset of $\mathbb{R}^m$.
Is it true that the zero set of $g(z)$ $$F_2 = \{z \in D \times V | g(z) = 0\}$$ has $(2n+m)$-dimensional Lebesgue measure?
How to prove ( or to modify the proof in this book, Corollary 10, p.9. to obtain) the above result?
