Example of non measurable function in $\mathbb R^2$

Question

I'm looking for a function $f:\mathbb R^2\to \mathbb R$ which is non-measurable. I have no ideas about this.

I know that if we have non-measurable sets in a measure space, there are non-measurable functions from that space.

Answer 1

Not every subset of $\mathbb R^{2}$ is a measurable set. If $A$ is a non-measurable set then define $f(x)=1$ for $ x \in A$ and $f(x)=0$ for $ x \in A^{c}$. Then $f$ is not measurable.

If $E$ is not measurable in $\mathbb R$ then $E \times \mathbb R$ is not measurable in $\mathbb R^{2}$. For an example of such a set $E$ see https://www.encyclopediaofmath.org/index.php/Non-measurable_set