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Ergodic Hypothesis

Assumes that, over long times, the time average of a system's property equals its ensemble average, a cornerstone of statistical mechanics.
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The statement of the theorem

Let (Γ,B,μ)(\Gamma, \mathcal{B}, \mu) be a measure space representing the phase space, where μ\mu is the invariant measure associated with the Hamiltonian flow x(t)\mathbf{x}(t). Let A:Γ→RA: \Gamma \to \mathbb{R} be a continuous observable function. The Ergodic Hypothesis asserts that for almost every initial point x0∈Γ\mathbf{x}_0 \in \Gamma (with respect to the measure μ\mu), the time average of AA equals the phase space average of AA: \n\nlim⁡T→∞1T∫0TA(x(t;x0))dt=∫ΓA(x)dμ(x)\lim_{T \to \infty} \frac{1}{T} \int_0^T A(\mathbf{x}(t; \mathbf{x}_0)) dt = \int_{\Gamma} A(\mathbf{x}) d\mu(\mathbf{x})\n\nThis equality holds provided the flow x(t)\mathbf{x}(t) is ergodic with respect to the measure μ\mu, meaning that for any measurable set E⊂ΓE \subset \Gamma such that x(t)∈E\mathbf{x}(t) \in E for almost all tt, the measure μ(E)\mu(E) must be either 0 or μ(Γ)\mu(\Gamma).
Source: Wikipedia