Example in Measure theory about Borel Measures

For a real number $x\in\mathbb R$ take $\mu_x(A)=\int_{A}\frac{\mathrm dt}{t^2}+x\delta_0(A)$

Then, $\mathbb R=\{0\}\cup \bigcup_n\left(]-\infty,-\frac1n[\cup]\frac1n,+\infty[\right)$ and each subset has finite measure so $\mu_x$ is $\sigma$-finite. Also it is clear that every $\mu_x$ has the same value on an open set which does not contain $0$, and is infinite as soon as $0$ is in the open set, so that all $\mu_x$ coincide on all open subsets but are not equal.

Thank you Kavi Rama Murthy for noticing me that my former answer was wrong.