TITLE: Is diameter of a set a measure?
QUESTION [8 upvotes]: Suppose the diameter of a nonempty set $A$ is defined as 
$$\sigma(A) := \sup_{x,y \in A} d(x,y)$$
where $d(x,y)$ is a metric.
Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable additivity for this particular case?

REPLY [4 votes]: Let $A$ be a circle of diameter 1, and let $B$ a circle of diameter 2, having the same center of $A$. Note that $A \subseteq B$.
Now, $\sigma(B) = 2$, yet $\sigma(B \setminus A) + \sigma(A) = 2 + 1 = 3$.