# Finite Borel measures that don't agree on R

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.

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Take $X$ to be the open intervals not containing $0$, one measure the zero measure and the other the Dirac point measure at $0$. – t.b. Nov 3 '11 at 16:41
I don't know if it may be helpful, but getting a finite measure is not so hard - in one way, you can use $\arctan$ to contract $\mathbb{R}$ to $(-\pi/2, \pi/2)$ and measure contracted sets or you can use integrals to define the measure of a set. Any integrable and non-negative function is OK. – savick01 Nov 3 '11 at 16:49