Originally Posted by

**hatsoff** The way I read this it's impossible. Let $\displaystyle f:E\to\mathbb{R}^n$ be a finite-valued Lebesgue measurable function. The codomain of $\displaystyle f$ is the finite set $\displaystyle \{y_1,\cdots,y_k\}$. We can put $\displaystyle \epsilon$-balls $\displaystyle B_i$ around each $\displaystyle y_i$ such that $\displaystyle B_1,\cdots,B_k$ are disjoint. Then $\displaystyle E_1,\cdots,E_k$ are measurable sets satisfying (1) and (2), where $\displaystyle E_i=f^{-1}(B_i)$.

Have I misunderstood something here?