The Impossibility Theorem

There can exist no measure $\mu : 2^{\mathbb {R}^{k}} \rightarrow \overline{\mathbb {R}}$ such that (i) $0 < \mu \left(\mathbb {R}^{k}\right) \leqslant +\infty$, (ii) $\mu$ assigns a finite nonnegative number to each bounded subset of $\mathbb {R}^{k}$, and (iii) $\mu \left(\left\{ \boldsymbol {x}\right\} \right) = 0$ for each $\boldsymbol {x} \in \mathbb {R}^{k}$. This is the $k$-dimensional version of the Impossibility Theorem.

Impossibility Theorem: There can exist no measure $\mu : 2^{\mathbb {R}} \rightarrow \overline{\mathbb {R}}$ with $\mu (\mathbb {R}) > 0$ that simultaneously (a) assigns finite measure to bounded sets, and (b) assigns zero measure to each singleton. In particular, Lebesgue measure cannot be consistently defined for every subset of $\mathbb {R}$.