 We are now in a position to introduce the Boltzmann entropy. To this end recall that we have a measure \(\mu \) on the phase space that assigns to every set a particular volume, hence a fortiori also to macrostates. With this in mind, the Boltzmann entropy of a macro-state \(M_j\) can be defined as \(S_B = k_B\log [\mu(M_j)]\), where \(k_B\) is the Boltzmann constant. The important feature of the logarithm is that it is a monotonic function: the larger \(M_j\), the larger its logarithm. From this it follows that the largest macro-state also has the highest entropy!