Mean of total energy as sum of means

Question

I have a question about the mean value of the total energy. I read that in the context of finding an adiabatic invariant for the pendulum \begin{equation} E = \langle T + V \rangle = \langle T \rangle + \langle V \rangle \end{equation} holds, with $E$ denoting the total energy, $T$ denoting kinetic energy and $V$ denoting potential energy. Which conditions are necessary that this is true, or is it valid in general?

Answer 1

Assume by default a joint distribution $ f_{TV}(t,v) $. Then, \begin{align} E(T + V) &= \int_t \int_v (t+v) f_{TV}(t,v) \, dv dt \\ &= \int_t \int_v t f_{TV}(t,v) \, dv dt + \int_v \int_t v f_{TV}(t,v) \, dt dv \\ &= \int_t t f_T (t) \, dt + \int_v v f_V (v) \, dv \\ &= E(T) + E(V) \end{align}