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The statement says almost everywhere because the equality of the time average and the ensemble average is asserted for all initial conditions except possibly a negligible exceptional set. The clip does not specify the exact nature of this exceptional set.

Conditions: The system is governed by an ODE $du/dt=f(u)$.; $J$ is a function of the state $u$.; The time average is defined by $\lim_{T\to\infty}\frac{1}{T}\int_0^T J(u(t))dt$.; The comparison ensemble has a density.

The statement says almost everywhere because the equality of the time average and the ensemble average is asserted for all initial conditions except possibly a negligible exceptional set. The clip does not specify the exact nature of this exceptional set.

Conditions: The system is governed by an ODE $du/dt=f(u)$.; $J$ is a function of the state $u$.; The time average is defined by $\lim_{T\to\infty}\frac{1}{T}\int_0^T J(u(t))dt$.; The comparison ensemble has a density.