A category-theoretic proof of the ergodic decomposition theorem

The ergodic decomposition theorem is a cornerstone result of dynamical systems and theory. It states that every invariant measure on system mixture ones. Here we formulate prove the in terms string diagrams, using formalism Markov categories. We recover usual measure-theoretic statement by instantiating our category stochastic kernels. Along way give conceptual treatment several concepts theory deterministic systems. In particular, - measures appear very naturally as particular cones morphisms (in sense categories); $\sigma$-algebra can be seen colimit line with other uses theory, once necessary structures are place, proof main much simpler than traditional approaches. it does not use any quantitative limiting arguments, rely cardinality group or monoid indexing dynamics. hope this paves for further applications to systems, information