Locally compact groups which are σ-compact are σ-finite under the Haar measure. For example, all connected, locally compact groups G are σ-compact. To see this, let V be a relatively compact, symmetric (that is V = V−1) open neighborhood of the identity. Then is an open subgroup of G. Therefore H is also closed since its complement is a union of open sets and by connectivity of G, must be G itself. Thus all connected Lie groups are σ-finite under Haar measure. Nonexamples Any non-trivial measure taking only the two values 0 and is clearly non σ-finite. One example in is: for all , if and only if A is not empty; another one is: for all , if and only if A is uncountable, 0 otherwise. Incidentally, both are translation-invariant. Properties The class of σ-finite measures has some very convenient properties; σ-finiteness can be compared in this respect to separability of topological spaces. Some theorems in analysis require σ-finiteness as a hypothesis. Usually, both the Radon–Nikodym theorem and Fubini's theorem are stated under an assumption of σ-finiteness on the measures involved. However, as shown in Segal's paper "Equivalences of measure spaces" (Am. J. Math. 73, 275 (1953)) they require only a weaker condition, namely localisability. Though measures which are not σ-finite are sometimes regarded as pathological, they do in fact occur quite naturally. For instance, if X is a metric space of Hausdorff dimension r, then all lower-dimensional Hausdorff measures are non-σ-finite if considered as measures on X. Equivalence to a probability measure Any σ-finite measure μ on a space X is equivalent to a probability measure on X: let Vn, n ∈ N, be a covering of X by pairwise disjoint measurable sets of finite μ-measure, and let wn, n ∈ N, be a sequence of positive numbers (weights) such that The measure ν defined by is then a probability measure on X with precisely the same null sets as μ. Related concepts Moderate measures A Borel measure (in the sense of a locally finite measure on the Borel -algebra) is called a moderate measure iff there are at most countably many open sets with for all and . Every moderate measure is a -finite measure, the converse is not true. Decomposable measures A measure is called a decomposable measure there are disjoint measurable sets with for all and . Note that for decomposable measures, there is no