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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 $0$.2011-11-03
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    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.2011-11-03
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    Now t.b. has an example where $X$ is a $\Pi$-system, but $\mu(\mathbb R) \ne \nu(\mathbb R)$. How about another example where $X$ is not a $\Pi$-system and $\mu(\mathbb R) = \nu(\mathbb R)$ ??2013-07-25

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