TITLE: Finite Borel measures that don't agree on $\mathbb{R}$
QUESTION [5 upvotes]: How would I go about finding a family $X$ of Borel sets in $\mathbb{R}$ that generate the Borel $\sigma$-algbera on $\mathbb{R}$ and two finite Borel measure $\mu$ and $\nu$ that agree on $X$ but do not agree on the whole Borel $\sigma$-algebra.
I know that $X$ cannot be a $\Pi$-system, so I was thinking of using the open intervals but I'm really struggling with the measures. The only finite measures I can think of are Dirac point measures.

REPLY [2 votes]: Is this question really 8 months old?  Try this:
Let $\mathcal X$ consist of all borel sets $A \subseteq \mathbb R$ such that 
$$
\text{either}\qquad
A \cap \{1,2,3,4\} = \{1,2\}\qquad\text{or}\qquad
A \cap \{1,2,3,4\} = \{1,3\} .
$$
Let $\mu = \delta_1+\delta_4$.  That is, points $1$ and $4$ each have measure $1$, everything else measure zero.  And let $\nu = \delta_2 + \delta_3$.  
Show: (a) $\mu(A)=\nu(A)$ for all $A \in \mathcal X$. (b) $\sigma(\mathcal X)$ is all Borel sets.