I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be welcomed. The question is as follows:
Suppose $\mu$ is a measure on $(\mathbb{R}, B(\mathbb{R}))$ such that $\mu((0,1])=1$ and $\mu$ is translation invariant. Here let $\lambda$ denote the Lebesgue measure.
(i) Show that for every interval $A = (a,b]$ with $b-a\in\mathbb{Q}$, we have $\mu(A)=\lambda(A)$.
(ii) show that $\mu=\lambda$
Like I said I think I can do the second part by showing the the sets in part one which the measure agree on are a $\pi$ system and then using the fact the Borel sets are generated by those intervals we can deduce the measure agree on the whole space, but I do not understand how to show they agree on the sets A in part (i). Thanks in advance!