I'm currently studying the book "Real Analysis for Graduate Students: Measure and Integration Theory" by Richard F. Bass, in page $17$, Exercise $9$ we have:
Suppose $X$ is the set of real numbers, $B$ is the set of Borel $\sigma$-algebra, and $m,n$ are two measures on $(X,B)$ such that $m((a,b))=n((a,b))< \infty$ whenever $-\infty <a<b< \infty$. Prove that $m(A)=m(A)$ whenever $A \in B$
I have proved the statement when $A$ is open, and my main problem arises as how can we categorize "Borel Measurable" sets, my first thought was that if $A$ is Borel measurable then $A- A^\circ$ is a null set of $B$ with respect to the natural measure ( i.e. $\mu ((a,b))=b-a$ ) but this doesn't seem to work, as there are fat Cantors.
I would like some explanation on what the "Borel measurable" really means and how can we (at least intuitively) identify them?
And as my second question, the problem seems to be implying that if for a measurable space $(X,A)$ there exists subsets of $X$ such as $\{E_i\}_{i \in I}$ where $A= \sigma(\{E_i\}_{i \in I})$ and $\{E_i\}_{i \in I}$ is $\sigma$-closed under intersection (the intersection of countably many $E_i$ is in $\{E_i\}_{i \in I}$ aswell) then if there exists measure functions $n,m$ such that $n(E_i)=m(E_i)< \infty$ for all $i \in I$ then $n(U)=m(U)$ for all $U \in A$, which indeed gives us a method of identifying measure functions without actually knowing it on all of $A$.
is my second question a correct generalization of the problem or Borel measurable sets and $\mathbb{R}$ are special, and if so how can we correctly generalize this problem for any measurable space?