Temporal Averages and Numerical Computations

We also note that lower order objects such as a point or a curve in or a surface in R, have zero measure with respect to Lebesgue measure for the ambient space. Similarly, for a Hamiltonian system with Nj degrees of freedom, the energy surface is of dimension 2Nd — 1 and the probability measure of sets of lower dimension than this will vanish. 

As we mentioned in motivating the microcanonical distribution, we could alternatively view microcanonical averages as integrals with respect to the singular measure on the ambient space 2) defined by 

Regardless of which formulation is adopted, we may use the probability measure to compute expectations (averages) of (suitable) functions defined on the phase space. We write such an expectation as 

Incidentally, any such function is referred to as an observable function (or, simply, an observable) of the system in this context. 

As shorthand, where the underlying space and probability density are assumed, one may write such an average more simply as 

---