- #1

- 138

- 0

Suppose ##X## is the set of real numbers, ##\mathcal B## is the Borel ##\sigma##-algebra, and ##m## and ##n## are two measures on ##(X, \mathcal B)## such that ##m((a, b))=n((a, b))< \infty## whenever ##−\infty<a<b<\infty##. Prove that ##m(A)=n(A)## whenever ##A\in \mathcal B##.

Here is what I am envisioning but I am not so sure: Since ##a, b \in \mathbb R## and since ##A## is an arbitrary subset of ##\mathcal B##, so if only I can prove that ##(a, b) \in \mathcal B##, then I am done. But here is my question:

How do I go ahead proving that ##(a, b) \in \mathcal B##? Can I just using the classic formula that if ##\forall a \in (a, b) \rightarrow a \in \mathcal B##, then ##(a, b) \in \mathcal B##? Any other step I need to follow?

Thanks for your time and effort.