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Math Help - real analysis...pls help me with this!

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    real 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) + [20 marks]
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    Quote Originally Posted by cynthiacheok View Post
    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) + [20 marks]
    The usual definition of outer measure m* is: m^*(E) = \inf\left\{\sum m(U_n)\right\}, where the infimum is taken over all countable collections of open intervals U_n such that E\subseteq\bigcup U_n.

    That means that we can find such a countable collection satisfying m^*(E) \leqslant \sum m(U_n) + \textstyle\frac14. Define U = \bigcup U_n. Then E ⊆ U. Also, m(U)\leqslant\sum m(U_n) and so m(E) \leqslant m(U) + \textstyle\frac14.
    Last edited by Opalg; August 23rd 2008 at 12:44 AM.
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