# Example in Measure theory about Borel Measures

I need to show that there are distinct $$\sigma$$-finite measures in $$B(\mathbb{R})$$ that are the same in the open sets. Im not really seeing a good example cause every open set is a countable union of compact sets, so the sum has to be the same , but they have to interchange themselves. Any help is appreciated.

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.

I think i found an example. If we consider $$\mu(A) = \# (\mathbb{Q} \cap A)$$ and $$\nu(A) = \# (A \cap(\mathbb{Q}\cup \sqrt{2}))$$.

• Are these measures $\sigma$-finite? May 13, 2019 at 12:41
• Yeah if we consider R to the union of the $\q_{ns}$ and the irrationals. May 13, 2019 at 12:42
• No they have 0 and 1 i think May 13, 2019 at 12:44
• Surely we have $\mu(\mathbb{Q}) = |\mathbb{Q}| = \infty$? May 13, 2019 at 12:45
• Yes we do , but we consider X to be the countable union of $q_{n}$ and each one of them has measure 1. May 13, 2019 at 18:37