The author constructs non-measurable sets. Let \(P\) be a locally compact Hausdorff space and let \((P, \mathfrak M_i, \mu_i)\) be a family of measure spaces such that for each \(i\), \(\mu_i\) is regular, \(\mathfrak M_i\) contains the family of Borel sets of \(P\), and \(\mu_i(\{x_i\}) = 0\) for each \(x\in P\). Then there exists \(T\subseteq P\), such that \(T\notin \cup_{i=1}^\infty \mathfrak M_i$. If \(P\) is compact, \(T\) can be so chosen such that for all $i\) the outer measure \({\mu_i}^*(T)\) of \(T\) is \(\mu_i(P)\) and the inner measure \({\mu_i}_*(T)\) of \(T\) vanishes. The basis of the construction is a result of F. Bernstein which states that there exists a subset \(S\) of the unit interval \(I$ such that any closed subset of \(I\) or of \(I-S\) has at most countably many elements. Let \(\mu\) be a regular measure on \(I\) and suppose \(S\) is measurable with respect to \(\mu\). If \(\mu(S)>0\), since \(\mu\) is regular, there exists a closed set \(H\subseteq S$ such that \(\mu(H)>0\). Since all closed subsets \(H\) of \(S\) are countable, \(\mu(S)\) must vanish. But, similarly, \(\mu(I-S)\) vanishes. A construction, based on that for Urysohn's Lemma, gives a mapping $\varphi$ from \(P\) into \(I\), such that for all \(i$, \(\mu_i\,[\varphi^{-1}(x)] = 0\) for all \(x\in I\). Define \(T= \varphi^{-1}(S)\).