$\mu$ is a measure on $(\mathbb{R}, B(\mathbb{R}))$ which is invariant by translation ($\forall a,b,h\in\mathbb{R}:\,\mu(]a+h, b+h]) = \mu]a, b]$) and such that $\mu(]0,1])=1$.
What I need to show is that $\mu(]0,x])=x$, for $x\in \mathbb{N}$, then for $x\in \mathbb{Q}_+$, then for $x\in \mathbb{R}_+$, and finally, to conclude that $\mu$ is the Lebesgue measure on $\mathbb{R}$.
Showing it for $x\in \mathbb{N}$ was fairly easy, but I am really struggling for the next steps. For $x\in \mathbb{Q}_+$, I tried to write $x$ as $x = \frac{p}{q}$ but it seems to lead nowhere, how are the cases $x\in \mathbb{Q}_+$ and $x\in \mathbb{R}_+$ different ?
Thanks :)
Let $p_n = \mu((0,1/n])$. Then by translation invariance, $p_n$ satisfies $$ 1 = \mu((0,1]) = np_n, $$ so $\mu((0,1/n])= 1/n$. From here, you should be able to say what the $\mu$-measure of an interval of the form $(0,m/n]$ is using translation invariance. To establish the measure of an arbitrary interval, use continuity of the measure $\mu$ and approximation of the endpoints by rational numbers.