  Using a Markov chain approach we rederive the exact density functional for
hard rod mixtures on a one-dimensional lattice, which forms the basis of the
lattice fundamental measure theory. The transition probability in the Markov
chain depends on a set of occupation numbers, which reflects the property of a
zero-dimensional cavity to hold at most one particle. For given mean occupation
numbers (density profile), an exact expression for the equilibrium distribution
of microstates is obtained, that means an expression for the unique external
potential that generates the density profile in equilibrium. By considering the
rod ends to fall onto lattice sites, the mixture is always additive.
