We almost can define a measure in a concise and neat way by stating that a measure on a $\sigma$-algebra $\Sigma$ is a countably additive function $ \mu : \Sigma \to [0, \infty] $ such that $ \mu(\emptyset) = 0 $. Except for the last property that I couldn't find a name for.
On my mind, such an essential should be given a name.
So far the best I could come up with is to call $\mu$ an empty-to-zero function.
References and ideas are welcome.
On
This is part of the definition of countable additivity.
Countable addivity applies to any countable collection of disjoint sets — in particular, it applies collections of size zero.
The union of a collection of sets of size zero is $\varnothing$, and the sum of a collection of numbers of size zero is $0$. Therefore, $\mu(\varnothing) = 0$.
Note that the above argument would not apply if you only assume addivity for countably infinite collections of sets, which is a weird thing to do, but it seems some really do use "countable additivity" in this more restrictive sense.
In this case, you can also invoke finite additivity (to which the above argument does apply), so that you have additivity for all countable cardinalities.
Additivity implies $\mu(\emptyset) = 0$.
$$\mu(\emptyset) = \mu(\emptyset \cup \emptyset) = \mu(\emptyset) +\mu(\emptyset) = 2 \mu(\emptyset), $$
with the second equality allowed because $\emptyset \cap \emptyset = \emptyset$.
It's all because of the weird properties of the empty set, so I think we include it to make the definition easier to think about, although it's not logically required.
Edit: Oops, $\mu(\emptyset)$ could also be $+\infty$ and satisfy that equation. I guess my argument doesn't work, though you could require "finite somewhere" and it would end up being equivalent to $\mu(\emptyset) = 0$, though it wouldn't be obvious.