Σ-finite measure

This can be generalized to more than two σ-finite measure spaces.

For instance, it is used in proving the existence claim of the Carathéodory extension theorem for σ-finite measures.

Conversely, any homogeneous system of imprimitivity is of this form, for some measure σ-finite measure μ.

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.

A set in a measure space is said to have σ-finite measure if it is a countable union of sets with finite measure.

The class of σ-finite measures has some very convenient properties; σ-finiteness can be compared in this respect to separability of topological spaces.

The algebra of all measurable subsets of a σ-finite measure space, modulo null sets, is a complete Boolean algebra.

The essentially bounded functions on a σ-finite measure space form a commutative (type I) von Neumann algebra acting on the L functions.

For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called non-commutative measure spaces.

In the following, we assume that μ is a σ-finite measure on a standard Borel G-space X such that the action of G respects the measure class of μ.

Tonelli's theorem states that on the product of two σ-finite measure spaces, a product measure integral can be evaluated by way of an iterated integral for nonnegative measurable functions, regardless of whether they have finite integral.