 Armed with this understanding of the definition of ergodicity, we can now discuss some important properties of ergodic systems. Consider a subset \(A\) of \(X\). For instance, thinking again about the example of the oscillating ball, take the left half of the phase space. Then define the so-called characteristic function of \(A, f_A\), as follows: \(f_A (x) = 1\) for all \(x\) in \(A\) and \(f_A (x) = 0\) for all \(x\) not in \(A\). Plugging this function into the definition of ergodicity yields: \(f^{*}_A = \mu(A)\). This means that the proportion of time that the system’s state spends in set \(A\) is proportional to the measure of that set. To make this even more intuitive, assume that the measure is normalised: \(\mu(X) = 1\) (this is a very common and unproblematic assumption). If we then choose \(A\) so that \(\mu(A) = 1\) ⁄ 2, then we know that the system spends half of the time in \(A\); if \(\mu(A) = 1\) ⁄ 4, it spends a quarter of the time in \(A\); etc. As we will see below, this property of ergodic systems plays a crucial role in certain approaches to statistical mechanics.