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Math Help - Limit of this counting measure

  1. #1
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    Limit of this counting measure

    Let  A \subset \mathbb {N} , define  \mu (A) = \lim _{n \rightarrow \infty } \frac {  count [A \cap \{ 1,...,n \} ] } {n}

    where "count" denote the number of elements in that set.

    Let M be the set of all even numbers, N be the set of all odd numbers, and P be the set of all perfect squares.

    Q1: Find  \mu (M) \ , \ \mu(N) \ , \ \mu (P)

    Q2: If A and B are disjoint sets, show that  \mu (A \cup B) = \mu (A) + \mu (B)

    Q3: Is it a measure?

    My solutions so far:

    Q1: For  \mu (M) = \lim _{n \rightarrow \infty } \frac { count [M \cap \{ 1,...,n \} } {n} , and I have  count [M \cap \{ 1,...,n \} = \frac {n}{2} since only half of those will be even, so the whole limit would equal to  \frac {1}{2} , same goes for all odd numbers.

    For the perfect squares, should it be  \frac { \sqrt {n} }{n} \rightarrow 0 ?

    And if I can prove Q2, doesn't that automatically prove Q3?

    Thank you!
    Last edited by tttcomrader; August 25th 2011 at 07:13 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Limit of this counting measure

    Quote Originally Posted by tttcomrader View Post
    Let  A \subset \mathbb {N} , define  \mu (A) = \lim _{n \rightarrow \infty } \frac {  count [A \cap \{ 1,...,n \} ] } {n}

    where "count" denote the number of elements in that set.

    Let M be the set of all even numbers, N be the set of all odd numbers, and P be the set of all perfect squares.

    Q1: Find  \mu (M) \ , \ \mu(N) \ , \ \mu (P)

    Q2: If A and B are disjoint sets, show that  \mu (A \cup B) = \mu (A) + \mu (B)

    Q3: Is it a measure?

    My solutions so far:

    Q1: For  \mu (M) = \lim _{n \rightarrow \infty } \frac { count [M \cap \{ 1,...,n \} } {n} , and I have  count [M \cap \{ 1,...,n \} = \frac {n}{2} since only half of those will be even, so the whole limit would equal to  \frac {1}{2} , same goes for all odd numbers.

    For the perfect squares, should it be  \frac { \sqrt {n} }{n} \rightarrow 0 ?

    And if I can prove Q2, doesn't that automatically prove Q3?

    Thank you!
    Q1's answer looks good. And, you tell us, does it? Are they the only two axioms an (outer) measure satisfies? Moreover, I hope it's clear how to prove Q2.
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