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**tttcomrader** Let $\displaystyle A \subset \mathbb {N} $, define $\displaystyle \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 $\displaystyle \mu (M) \ , \ \mu(N) \ , \ \mu (P) $

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

Q3: Is it a measure?

My solutions so far:

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

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

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

Thank you!