# Invariant measures

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If is a map then a measure is **invariant** under if for all . In the context of Hamiltonian flows, an **invariant measure** on phase space is invariant under the Hamiltonian flow map. This measure is a probability measure if it is positive and has total measure of size 1.

## Examples of invariant measures

- Gibbs measure
- Every invariant torus supports at least one invariant measure.