MHB Is it enough to show that it is a Lebesgue measure?

[mathmari]

Hey!

I want to show that when $\mu$ is a Borel measure in $\mathbb{R}$ with $\mu([0,1))=1$, which is a translation invariant, then it is also a Lebesgue measure.

I have shown that $\mu([a,b))=b-a, \forall a,b \in \mathbb{Q}$.

Is it enough to show that $\mu$ is a Lebesgue measure?? (Wondering)

 

[ThePerfectHacker]

If two Borel measures agree on intervals of the form $[a,b)$ then they agree on anything in the $\sigma$-algebra which is generated by those intervals.