  We present an approach to the study of stationary measures placing Tarski's
foundational work in this area within a modern category theoretic context.
Guiding this work is the notion that measurable spaces equipped with symmetries
carry an intrinsic notion of `size'. We demonstrate that within such measurable
spaces, the collection of these sizes embeds into a partially ordered
commutative inverse monoid. Using this we prove, in great generality, the basic
theorems of measure theory and give an algebraic characterization of the space
of classical real valued stationary measures.
