(Measure theory, functional analysis:)

Let (μ, X, **B**) be a measure space on a set X. As expected, the space is called *finite* if μ(X)<∞. (**NOTE** that the set X needn't be finite at all!) The space is called *σ-finite* (yes, that's a Greek letter participating in spelling an English word; mathematicians are weird that way...) if there exists a countable partition X=X_{1}∪X_{2}∪... (of X into measurable subsets X_{n}∈**B**) such that μ(X_{n})<∞ for all n. That is, a measure space is σ-finite iff it is a countable union of finite measure spaces.

Often, just the measure μ will be called "σ-finite", when there can be no confusion regarding X and **B**.

For example, Lebesgue measure m on **R**^{d} is *not* a finite measure. However, m *is* σ-finite. You can see this e.g. by splitting **R**^{d} into countably many cubes X_{n}.

Many results of functional analysis and measure theory don't hold for just *any* measure space, but can be shown for finite measure spaces. But real analysts are much more interested in **R**^{d}, which is not finite; by extending the results to σ-finite spaces (something which is often rather easy to do, by utilizing the "continuity" of σ-additive measures) we can show them for the spaces of interest, such as **R**^{d}.