Lebesgue measures are sigma finite

Question

How can I show a Lebesgue measure on $\mathbb{R}$ is $\sigma$-finite? I know that a measure $\mu$ on $(\mathbb{R},\mathfrak{B}(\mathbb{R}))$is a Lebesgue measure on $R$ if $\mu (A)$ is the length of A for every interval of A. But how do I show such a measure is $\sigma$ finite?

Answer 1

Remember that a space is $\sigma$-finite if it is the countable union of $\mu$-finite subsets. We can write $\mathbb R$ as a countable union of increasing intervals $$ \mathbb R = \bigcup_{k=1}^\infty [-k, k]. $$ And we know $\mu([-k,k]) = 2k < \infty$.