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Math Help - zero measure

  1. #1
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    zero measure

    How do you show that any subset of a zero measure set has measure zero?
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  2. #2
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    Quote Originally Posted by vinnie100 View Post
    How do you show that any subset of a zero measure set has measure zero?
    Is the measure monotone?
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  3. #3
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    Quote Originally Posted by vinnie100 View Post
    How do you show that any subset of a zero measure set has measure zero?
    Well, if we're talking Lebesgue measure...

    Let Z be a zero set and A be a subset of Z

    we know:

    m*(A) >= 0 (by definition)

    and m*(A) <= m*(Z) = 0, this follows by monotonicity

    so 0 <= m*(A) <= 0

    and m*(A) = 0
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  4. #4
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    We defined a set to have measure zero if for any epsilon>0, there is (at most countable) collection of intervals that cover A and whose total length is less than epsilon.

    Hint: For A_n, choose a cover with epsilon_n = epsilon/(2^n)
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  5. #5
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    Quote Originally Posted by vinnie100 View Post
    We defined a set to have measure zero if for any epsilon>0, there is (at most countable) collection of intervals that cover A and whose total length is less than epsilon.

    Hint: For A_n, choose a cover with epsilon_n = epsilon/(2^n)
    Why wouldn't the cover for the superset work for the subset?
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