Holomorphic function of several complex variables and sets of zero Lebesgue measure

Question

Let $f$ be a holomorphic function of $n$ complex variables in region $\Omega \subset \mathbb{C}^n$, with $\Omega \supset \mathbb{R}^n$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^n$. Is the following statement true?

For any set $A \subset \mathbb{R}^n$ with $\lambda(A) = 0$, $$ \lambda\left( f^{-1}(f(A)) \cap \mathbb{R}^n \right) = 0 . $$