 The importance of this is that it allows us to calculate expectation values. Assume that the game is such that you get one Pound if the dart hits the left half and three Pounds if it lands on the right half. What is your expected gain? The answer is 1 ⁄ \(2 \times 1\) Pound \(+ 1\) ⁄ \(2 \times 3\) Pounds \(= 2\) Pounds. This is the expectation value. The same idea is at work in SM. Physical magnitudes like, for instance, pressure are associated with functions \(f\) on the phase pace. We then calculate the expectation value of these magnitudes, which, in general is given by \(\langle f \rangle = \int fd\mu\). In the context of Gibbsian SM these expectation values are also referred to as phase averages or ensemble averages. They are of central importance because these values are used as predictions for observed values. So if you want to use the formalism to predict what will be observed in an experiment, you first have to figure out what the probability density \(\varrho\) is, then find the function \(f\) corresponding to the physical quantity you are interested in, and then calculate the phase average. Neither of these steps is easy in practice and working physicists spend most of their time doing these calculations. However, these difficulties need not occupy us if we are interested in the conceptual issues underlying this ‘recipe’.