semifinite

Semifinite is a term used in measure theory and operator algebras to describe a relaxation of finiteness conditions. In measure theory, a measure μ on a measurable space (X, Σ) is semifinite if every measurable set of infinite measure contains a subset of finite positive measure. Equivalently, whenever μ(A) = ∞ there exists B ⊆ A with 0 < μ(B) < ∞. Semifiniteness ensures that one can locate nontrivial finite pieces inside any “large” set, which is important for developing integration theories in general contexts.

Relation to sigma-finiteness: A measure that is sigma-finite is automatically semifinite, because X can be written as a countable union of finite-measure sets, each providing finite pieces. The converse is not true in general. A standard example of a semifinite but not sigma-finite measure is the counting measure on an uncountable set: every infinite subset has finite nonzero subsets, but the space cannot be covered by countably many finite-measure pieces.

In operator algebras, semifiniteness also appears in the notion of traces. A trace τ on a von Neumann algebra M is semifinite if for every positive element a with τ(a) = ∞ there exists a nonzero projection p ≤ a with 0 < τ(p) < ∞. A von Neumann algebra is called semifinite if it admits a faithful normal semifinite trace. Commutative examples arise from semifinite measures: if μ is a semifinite measure on (X, Σ), then the algebra L∞(X, μ) carries a faithful normal semifinite trace given by τ(f) = ∫ f dμ.

Applications of semifiniteness include the construction of noncommutative Lp-spaces and the study of integration in settings that are not globally finite or sigma-finite.