Is diameter of a set a measure?

Observe that the diameter of singletons is $0$ and the diameter of set $\{x,y\}$ is $d(x,y)>0$ if $x\neq y$. So there is no additivity.


It's not even finitely additive. If $X$ and $Y$ are two disjoint closed intervals on the real line then the diameter of their union is not the sum of their diameters.


... not to mention

$\sigma( \text{rational numbers between A and B}) + \sigma( \text {irrational numbers between A and B}) \ne \sigma( \text{ real numbers between A and B})$.

This is pretty much the perfect example of something that absolutely can not be a measure and illustrates why we need a concept of measure.

Tags:

Metric Spaces

Real Analysis

Measure Theory

Lebesgue Measure

Functional Analysis

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