The construction of a Lebesgue measurable set with every density (Q802776)

The author gives the construction of a Lebesgue measurable set corresponding to every density \(t\in [0,1].\) In fact, he proves the following Proposition: Given \(0\leq \alpha \leq 1,\epsilon >0\) and \((a,b),\) there exists a measurable set \(A\subset (a,b)\) such that \(\lambda A=\alpha (b-a)\) and for every \(c\in (a,b),\) \[ | \frac{\lambda (A\cap (a,c))}{c-a}-\alpha | <\epsilon \text{ and } | \frac{\lambda (A\cap (c,b))}{b-c}-\alpha | <\epsilon. \]