Question on measurable set I am having a hard time solving the following problem.  Any help would be wonderful.
If $f: [0, 1] \rightarrow [0, \infty)$ is measurable, and we have that $\int_{[0, 1]}f \mathrm{d} m = 1$, must there exist a continuous function $g: [0, 1] \rightarrow [0, \infty)$ and a measurable set $E$ with $m(E) = \frac{3}{4}$ and $|f - g| < \frac{1}{100}$ on $E$.

$m([f > 10]) \le 1/10$ (Chebyshev)
Uniformly approximate $f$ on $[0 < f \le 10]$ by a simple function, within say 1/200.
By linearity, reduce to characteristic function of a measurable set. But a measurable set is close to a finite union of intervals (the measure of their symmetric difference can be made small). 
