To my understanding, the Haar-measure on a locally compact group $G$ is inner regular on all open subsets and compact subsets have finite measure. Why then would the group itself, being open, not be $\sigma$-finite? Couldn't we find a sequence of compact subsets, all with finite measure, whose measure converges (or diverges to infinity) to the measure of the group, making it $\sigma$-finite?
If it turns out that not all locally compact groups are $\sigma$-finite, does fubini's theorem still hold?