I was reading Sheldon Axler's Real Analysis book and he mentions the following (as does the course I am taking) :
There does not exists a function $\mu : \mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ with the following properties:
- $\mu( (a,b) ) = b-a$
- $\mu (\underset{n\in \mathbb{Z}_{+}}{\bigcup} A_n) = \underset{n\in \mathbb{Z}_{+}}{\sum} \mu (A_n)\quad $whenever $A_n$'s are pairwise disjoint sets.
- $\mu(t+A) = \mu(A)$ for any $t\in \mathbb{R}, A \subset \mathbb{R}$.
I understand the proof of this theorem but somehow translation invariance seems additional in the sense that whatever function is induced by the first two properties, must have the third one since $\mu((t+a, t+b)) = b-a$
Then is there a map $\mu$ with only the first two properties? (so that third condition really put restrictions for the choice of $\mu$)