An example of a non $\mu^*$-measurable set under the projection function

Question

I have this problem and I am just stuck. I'd appreciate any help. Thanks.

Let $\lambda^*$ be the Lebesgue outer measure on $\mathbb{R}$ and $\pi_x(x,y)=x$, be the projection onto the x-axis. Define a function $\mu^*: \mathfrak{B}(\mathbb{R}^2)\rightarrow[0,\infty]$ by

$$\mu^*(B)=\lambda^*(\pi_x(B))$$

Find a simple example of a set $B\subset \mathbb{R}^2$ that is not $\mu^*$-measurable.

I know I need to show $\mu^*(A)\neq\mu^*(A\cap B)+\mu^*(A\cap B^\complement)$ but I don't really know how to choose $B$.

Answer 1

Let $E$ be a non-measurable set in $\mathbb R$ so that $\lambda^{*}(A) \neq \lambda^{*}(A \cap E)+\lambda^{*}(A\cap E^{c})$ for some $A$. Let $B=E\times \mathbb R$. If $C=A\times \mathbb R$ then $\mu ^{*}(C) \neq \mu^{*}(C \cap B)+\mu^{*}(C\cap B^{c})$, so $B$ is not $\mu^{*}$ measurable.