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**davidmccormick** Let $\displaystyle f(x) = x^{-1/2}$ if 0 < x < 1, $\displaystyle f(x) = 0$ otherwise. Let $\displaystyle \{r_n\}$ be an enumeration of rationals, and define $\displaystyle g(x) = \sum (1/2^n)f(x-r_n)$. Show that g is discontinuous at every point and unbounded on every interval.

Show that if f is a non-negative integrable function, for every $\displaystyle \epsilon > 0$ there exists $\displaystyle E \in M$ such that $\displaystyle m(E) < \infty$ and $\displaystyle \int_{E} f > \int f - \epsilon$