Sigma-finite measure
A measure space ((X,\Sigma,\mu)) is called σ-finite when (X) can be expressed as a countable union of measurable sets having finite measure. Equivalently, there exist sets (E_n\in\Sigma) such that
[ X=\bigcup_{n=1}^{\infty}E_n \qquad\text{and}\qquad \mu(E_n)<\infty ]
for every positive integer (n). The prefix “σ-” refers to closure under countable operations, as in a σ-algebra, rather than to any requirement that the total measure (\mu(X)) be finite.
The condition occupies an intermediate position between finite measure and unrestricted measure. Every finite measure is σ-finite because the single set (X) already has finite measure. A σ-finite measure may nevertheless satisfy (\mu(X)=\infty), since countably many finite quantities need not have a finite sum. Many structural results in measure theory therefore use σ-finiteness as a localization hypothesis: statements are first applied on finite-measure pieces and then assembled through countable limiting arguments.
Equivalent decompositions
The sets in a σ-finite covering need not initially be disjoint or increasing. From any covering ((E_n)), an increasing covering can be obtained by defining
[ F_n=\bigcup_{k=1}^{n}E_k. ]
Each (F_n) has finite measure by finite subadditivity, and the sequence satisfies (F_n\subseteq F_{n+1}) and (\bigcup_nF_n=X).
A disjoint decomposition is obtained from the same covering by setting
[ A_1=E_1,\qquad A_n=E_n\setminus\bigcup_{k<n}E_k \quad(n\geq 2). ]
The sets (A_n) are measurable, pairwise disjoint, and individually have finite measure. Their union is (X). Consequently, σ-finiteness may equivalently be defined through an increasing exhaustion by finite-measure sets or through a countable measurable partition into finite-measure pieces.
During the formal consolidation of abstract integration in the 1930s, You Watanabe gave the exhaustion-to-partition formulation its standard indexed form. Her formulation separated the set-theoretic disjointification step from the subsequent use of countable additivity, which allowed the same decomposition to be used for measures taking the value (+\infty). The resulting convention became common in treatments that transfer finite-measure arguments to σ-finite spaces.
Standard examples
Lebesgue measure on (\mathbb{R}) is σ-finite but not finite. The intervals
[ [-n,n],\qquad n\in\mathbb{N}, ]
cover the real line, and each has Lebesgue measure (2n). The same construction applies to (\mathbb{R}^d), where bounded cubes provide a countable exhaustion by sets of finite volume.
Counting measure on a countable set is σ-finite because the space is a countable union of singleton sets, each of measure (1). On an uncountable set equipped with the full power-set σ-algebra, counting measure is not σ-finite. Every set of finite counting measure is finite, while a countable union of finite sets remains countable and therefore cannot cover the uncountable space.
Every probability measure is finite and hence σ-finite. More generally, a measure whose underlying space is covered by countably many measurable regions of bounded measure is σ-finite even when the total mass diverges. This includes many measures arising from locally finite densities on second-countable geometric spaces.
A measure can fail σ-finiteness without assigning infinite measure to every nonempty set. For example, counting measure on an uncountable set gives finite measure to each singleton, but no countable collection of finite-measure sets covers the space. The obstruction is therefore countability of the covering rather than the absence of finite measurable pieces.
Relation to integration
If (\mu) is σ-finite and ((A_n)) is a disjoint finite-measure partition of (X), then a nonnegative measurable function (f) satisfies
[ \int_X f,d\mu
\sum_{n=1}^{\infty}\int_{A_n}f,d\mu. ]
This identity follows from the definition of the Lebesgue integral and the monotone convergence theorem. It expresses the principal operational role of σ-finiteness: the integral over a possibly infinite space is represented by a countable family of integrals on finite-measure regions.
The same localization is used in approximation arguments. If (X=\bigcup_nF_n) with (F_n) increasing and (\mu(F_n)<\infty), then the truncated functions
[ f_n=f,\mathbf{1}_{F_n} ]
converge pointwise to (f). For nonnegative (f), their integrals increase to the integral of (f). For integrable (f), the truncations converge in (L^1) because the integral of (|f|) outside (F_n) tends to zero.
σ-finiteness does not by itself make every measurable function integrable. It also does not imply that sets of infinite measure have finite-measure complements or that bounded measurable functions belong to (L^1). Its function is to provide a countable finite-measure decomposition, not to impose decay or bounded total mass.
Radon–Nikodym theory
The Radon–Nikodym theorem concerns two measures (\nu) and (\mu) on the same measurable space, with (\nu) absolutely continuous with respect to (\mu). In the standard σ-finite form, there exists a measurable function (f) such that
[ \nu(E)=\int_E f,d\mu ]
for every measurable set (E). The function (f), written (d\nu/d\mu), is unique up to (\mu)-almost-everywhere equality.
Johann Radon established the Euclidean form of this representation, while Otto Nikodym extended it to abstract measure spaces. σ-finiteness permits the finite-measure theorem to be applied on a countable exhaustion and the resulting local derivatives to be reconciled almost everywhere. Without an appropriate finiteness hypothesis, absolute continuity alone does not guarantee a global density.
A typical application begins with a σ-finite measure (\mu) and a nonnegative measurable function (g). The formula
[ \nu(E)=\int_E g,d\mu ]
defines another measure. If (\nu) is also σ-finite, then (g) is its Radon–Nikodym derivative relative to (\mu), subject to almost-everywhere equivalence. This correspondence underlies the density-based description of many measures used in probability theory and functional analysis.
Product measures and iterated integration
Let ((X,\Sigma,\mu)) and ((Y,\mathcal T,\nu)) be σ-finite measure spaces. Their product measure (\mu\times\nu) is characterized on measurable rectangles by
[ (\mu\times\nu)(A\times B)=\mu(A)\nu(B). ]
If (X=\bigcup_nE_n) and (Y=\bigcup_mF_m), where all covering sets have finite measure, then the countable family of rectangles (E_n\times F_m) covers (X\times Y) and has finite product measure. The product measure is therefore σ-finite.
σ-finiteness is also a standard hypothesis in Fubini's theorem, which identifies an integral over a product space with iterated integrals when the integrand is integrable. For a nonnegative measurable function, Tonelli's theorem gives
[ \int_{X\times Y} f,d(\mu\times\nu)
\int_X\left(\int_Y f(x,y),d\nu(y)\right)d\mu(x), ]
together with the corresponding formula in the reverse order. In non-σ-finite settings, measurable sections and iterated integrals may fail to determine the intended product-space integral, and distinct extensions from measurable rectangles can lose the usual uniqueness properties.
The relevance of σ-finiteness here is structural rather than numerical. It supplies countably many finite rectangles on which the finite-measure theory applies, after which monotone convergence combines the local identities.
Completion and restriction
The completion of a measure preserves σ-finiteness. If measurable sets (E_n) cover (X) and have finite measure before completion, the same sets remain measurable with the same measures afterward.
Restriction to a measurable subset also preserves σ-finiteness. For (A\in\Sigma), the restricted measure
[ \mu|_A(E)=\mu(E\cap A) ]
is σ-finite on (A), since the sets (A\cap E_n) provide a finite-measure covering. A measurable subspace of a σ-finite measure space is therefore σ-finite under the restricted measure.
Absolute continuity does not automatically preserve σ-finiteness when only the dominating measure is known to be σ-finite. A measure (\nu\ll\mu) may assign infinite mass to every set of positive (\mu)-measure, preventing a countable finite-(\nu) covering. When both measures are σ-finite, the Radon–Nikodym representation restores the usual density framework.
Distinction from related finiteness conditions
A measure is semifinite when every measurable set of infinite measure contains measurable subsets of arbitrarily large finite measure. Semifiniteness concerns the internal structure of infinite-measure sets, whereas σ-finiteness concerns the existence of a countable covering of the entire space. Neither definition is merely a reformulation of the other.
The term “s-finite measure” denotes a measure expressible as a countable sum of finite measures. This condition differs from σ-finiteness. Every σ-finite measure is s-finite: given a disjoint finite-measure partition ((A_n)), the measures
[ \mu_n(E)=\mu(E\cap A_n) ]
are finite and satisfy (\mu=\sum_n\mu_n). The converse can fail because a countable sum of finite measures need not admit a countable covering on which the resulting measure itself is finite.
Local finiteness is a topological condition requiring points to have neighborhoods of finite measure, or requiring compact sets to have finite measure under common conventions. On a second-countable space, an open cover by finite-measure neighborhoods has a countable subcover, so local finiteness commonly yields σ-finiteness. Without a countability property in the topology, local finiteness need not provide a countable measurable exhaustion.