$\left\{x:f(x)\neq 0\right\}$ decomposes into a numerable union of sets of finite measurement. with $f$ complex valued function and integrable.

Question

Let $f$ function with valued in $\mathbb{C}$ and $f\in L^1(\mathbb{R})$. The set $\left\{x:f(x)\neq 0\right\}$ decomposes into a numerable union of sets of Finite measurement.

I want $\left\{x:f(x)\neq 0\right\}=\bigcup_{n\in \mathbb{N}} \left\{x:|f(x)|>1/n\right\} $ But I do not know if this applies in this case with complex values ​​...