I am working on this problem on measure theory like this:
Let $X$ be set of $\mathbb R$, and let $\mathcal B$ be its Borel $\sigma$-algebra, and finally let $\mu_1$ and $\mu_2$ be the two measures on $(X,\mathcal B)$ such that $\mu_1((a,b))= \mu_2((a,b)) < \infty$ whenever $−\infty < a < b < \infty$. Show that $\mu_1(A) = \mu_2(A)$ whenever $A \in \mathcal B$.
Here is what I was at first thinking: 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 I was told by a responder to my posting at Physics Forum here that this reasoning is wrong, since not all sets in $\mathcal B$ are open. I am hitting a deadend again.
Therefore I am posting this question here looking for help, thanks for your time and effort.
POST SCRIPT - 1: I should have mentioned this: This problem comes from the 3rd. chapter of an introductory text by Richard F. Bass here, therefore any solution shouldn't involve any advanced theorems such as Dynkin's, etc. Sorry for this belated info, thanks though to all who have taken time to help.
POST SCRIPT - 2: I finally came up with solution without any advanced theorems, adapted from a solution by @JoshKeneda, who used Dynkin's Theorem. I have submitted this work to my professor, he ok'd it except for (5) because it is true only when the $A_i$'s are pairwise disjoint. Feel free to drop me a message if you have ideas to improve (5). Thanks to all and especially to @JoshKeneda.
DISCLOSURE: This question is very similar to an old MSE posting here, which was put on hold due to being incomplete. My posting has all the correction to the first posting. Always conscientious of community rule and guideline, I have tried avoiding duplication by posting this question elsewhere here and here, but I did not receive any meaningful helps $-$ understandably, as those two outside forums are not specialized in math. Thank you for your understanding.