1. Let a set E have a finite outer measure. Show that if E is not measurable, then there is an open set O containing the set that has finite outer measure and for which m*(O~E) > m*(O)-m*(E).
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A set is measurable if and only if both outer and inner measure are finite and equal. The inner measure is non-negative and less than the outer measure so since the outer measure is finite, so is the inner measure. That means that, since this set is not measurable, the inner measure must be less than the outer measure. What if O is an open set, containing E, that is measurable?