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Math Help - borel set

  1. #1
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    borel set

    Show that every lebesgue measurable subset of R is the union of a Borel subset of R and a set of of outer measure zero.
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    Show that every lebesgue measurable subset of R is the union of a Borel subset of R and a set of of outer measure zero.
    Let E be a lebesgue measurable subset of R

    For n = 1,2,3,... we can find a set of open intervals {  I^n_k} suvh that  \sum_{k=1} {m(I^n_k)} \leq  m(E) + 1/n

    Let  G_n = \bigcup_{k=1}{I^n_k} . Then  G_n is borel since it is the union of open sets.

    Let  G = \cap{G_n} , then G is also Borel.

    Finally, I claim m(G\E) = 0, it suffices to show m(E) = m(G)

    Since E is in G, then m(E) <= m(G)

    And,  m(G) \leq m(G_n) \leq m(E) + 1/n

    And this is true for all n, thus m(G) <= m(E), and so m(E) = m(G)

    and m(G/E) = 0
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