Let E be a nonempty subset of R. Find an open set U of R that satisfies the following two conditions: E C U and m*(U) ≤m*(E) + ¼ [20 marks]

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- Aug 22nd 2008, 02:36 PMcynthiacheokreal analysis...pls help me with this!
Let E be a nonempty subset of R. Find an open set U of R that satisfies the following two conditions: E C U and m*(U) ≤m*(E) + ¼ [20 marks]

- Aug 23rd 2008, 12:05 AMOpalg
The usual definition of outer measure m* is: $\displaystyle m^*(E) = \inf\left\{\sum m(U_n)\right\}$, where the infimum is taken over all countable collections of open intervals $\displaystyle U_n$ such that $\displaystyle E\subseteq\bigcup U_n$.

That means that we can find such a countable collection satisfying $\displaystyle m^*(E) \leqslant \sum m(U_n) + \textstyle\frac14$. Define $\displaystyle U = \bigcup U_n$. Then E ⊆ U. Also, $\displaystyle m(U)\leqslant\sum m(U_n)$ and so $\displaystyle m(E) \leqslant m(U) + \textstyle\frac14$.