Problem. Construct a function in which is not in for any .
Solution. Let The particular thing to note about this function is that but if then (The exact details of can be varied; all we really need are those two properties). Let be an enumeration of the rational numbers (any other countable dense subset of would do just as well). Let
(The interchange of integral and sum is justified by the Monotone Convergence Theorem since is nonnegative).
But on any interval chose some On that interval,