TITLE: Can the graph of a non-measurable function be a zero set?
QUESTION [0 upvotes]: If so, what is an example of a non-measurable function in R that has a zero graph (graph is a zero set)? Thank you!

REPLY [2 votes]: Let $A\subseteq \mathbb{R}$ be any non-measurable set. Let $f\colon \mathbb{R}\to \mathbb{R}$ be the indicator function $$f(x) = \begin{cases}1 & \text{if }x\in A\\ 0&\text{if }x\notin A\end{cases}.$$
Then $f$ is not measurable. But the graph of $f$ is contained in the null set $$\{(x,0)\mid x\in \mathbb{R}\}\cup \{(x,1)\mid x\in \mathbb{R}\},$$ so it is a null set too.
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