Important examples of measures which are not $\sigma$-finite

Question

I think a measure which is not $\sigma$-finite is pain in the ass. I wish I could safely assume all the measures are $\sigma$-finite. I wonder if my wish is reasonable. Here's my question: what are important examples of measures which are not $\sigma$-finite? Of course, "important" is a subjective word. So this is a soft question. This is also a big-list question.

Answer 1

Counting measure on an uncountable set is not $\sigma$-finite.

Answer 2

A non trivial measure taking only the values $0$ and $\infty$ is non $\sigma$-finite .

Answer 3

I believe the most important class of non-$\sigma$-finite measures is provided by the Hausdorff measures. $\mathcal{H}_{\alpha}$ on $\mathbb{R}^d$ where $d > \alpha$ is not $\sigma$-finite. (Because any subset of finite $\mathcal{H}_{\alpha} $ measure is of null Lebesgue measure)

Answer 4

Consider ${\Bbb R}^2$ with a $\sigma$-algebra $\Sigma$ generated by smooth curves of finite length. Construct a measure $\mu$ so that $\mu(\gamma) = $length of $\gamma$ if $\gamma$ is a curve. This is a measure which is not $\sigma$-finite. The importance of this example is that the map $\mu:\gamma\mapsto\int_{\gamma}d\ell$ is a natural consideration as a measure for an interesting kind of subset (curves) in ${\Bbb R}^2$. In general, on any $n$-dimension manifold, the measure $A\mapsto \int_{A}\omega$ is not $\sigma$-finite if $\omega$ is a differential form or a pseudo differential form of degree $k<n$.