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

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.

• Most people would avoid measures that are not $\sigma$-finite as many theorems fail miserably. A former advisor of mine said that he thinks a version of Fubini-Tonelli holds in non-$\sigma$-finite spaces (and seemingly had a proof) but he was not entirely convinced by it. Non-$\sigma$-finite measures are very hard to conceive (outside of a few examples) since nearly everything we deal with across swaths of mathematics is $\sigma$-finite. I suggest turning this into a Community Wiki post because it could prove to be very helpful and big list questions usually are Community Wiki. Oct 1 '14 at 1:33
• @CameronWilliams I've read several times that Fubini theorems remain valid for s-finite measures, measures that are countable sums of finite measures. May 7 '21 at 23:59

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 $k<n$.

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)

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

• I know. Could you explain why you think it is important?
– user179340
Oct 1 '14 at 1:27

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

• Could you explain why you think it is important?
– user179340
Oct 1 '14 at 1:30
• it is useful to keep in mind if you are looking for easy counterexamples, which is all I ever did with this measure :D
– user67133
Oct 1 '14 at 1:32