Decomposable measure

In mathematics, a decomposable measure is a special kind of measure that can be thought of as a collection of smaller, simpler measures put together without overlapping. It is made by combining finite measures in a way that they do not share any parts. This idea builds on another concept called σ-finite measures, which are also made from smaller parts, but only from a countable number of them. Many important results in measure theory, like the Radon–Nikodym theorem, work for σ-finite measures but not for all measures. These results also work for decomposable measures, which are a broader group. Even though decomposable measures are more general, most of the ones we actually use in real problems are still σ-finite.

Examples

The counting measure on an uncountable space where every subset can be measured is a decomposable measure, but it is not σ-finite. Important theorems like Fubini's theorem and Tonelli's theorem work for σ-finite measures but might not work for this one.

The counting measure on an uncountable space where not every subset can be measured is usually not a decomposable measure. Also, a space with just one point and infinite measure is not decomposable.