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Trying to find the measure of a set.

  1. Feb 7, 2010 #1
    I am trying to learn about Lebesgue measure. One of the questions I couldn't solve is this;

    Show that the following sets are Lebesgue measurable and determine their measure

    A = {x in [0,1) : the nth digit in the decimal expansion is equal to 7}

    B = {x in [0,1) : all but finitely many digits in the decimal expansion are equal to 7}

    Now, the book defines a set E to be Lebesgue measurable if E = A U B, where A is in the Borel $\sigma$-algebra and B is a null set (outer measure 0), but I don't see where that helps here. Any hints?
  2. jcsd
  3. Feb 8, 2010 #2
    Ok, I figured out A, but I'm not sure on B. Any help?
  4. Feb 8, 2010 #3


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    The measure is 0. The general idea is that for any n if n digits are 7, then the measure is (1/10)n. So let n -> ∞
  5. Feb 8, 2010 #4
    Yes, but that doesn't prove that it's measurable. To prove its so, you have to write it as A U B, A borel and B null. Then the measure part I get.
  6. Feb 9, 2010 #5


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    If its null it is measurable. You can always union it with the empty set of you insist on having a Borel set to union with.
  7. Feb 10, 2010 #6
    For a fixed finite set of decimal place not equal to seven the number of points is finite. So for instance the number of points if only the first 2 place are not 7 is 100. The number of finite subsets of a countable set is countable, I think. So the union of all of these finite sets is countable and thus Borel measurable zero.
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