- #1
ForMyThunder
- 149
- 0
In Elias Stein's book Real Analysis, a measurable set [tex]E[/tex] is a set such that for every [tex]\epsilon>0[/tex], there exists an open [tex]\mathscr O [/tex] with the property that [tex]m_*(\mathscr{O}-E) < \epsilon[/tex]. But for every open set that covers the rationals in, say, [tex] [0,1] [/tex] must cover the entire interval so that the set of rationals can't satisfy the conditions for a measurable set. But then, this set of rationals in the unit interval is the countable union of point sets so it MUST BE measurable. Where was the hole in my argument that the set isn't measurable?