The following is split into two chapters. The first chapter gives a brief history concerning g-measures, their state of investigation and under what conditions, on g, unique g-measures exist. It concludes by giving equivalent conditions for a g-function to have a unique g-measure. This will, possibly, lead to a solution to Keane’s original problem about the uniqueness of a g-measure for an arbitrary g-function. The second chapter generalises the result of Prof. K. Schmidt that the Beta-function is invariant under finitarily isomorphic (with finite expected code length) Markov spaces, to g-spaces with certain conditions on the g-function. The approach adopted is essentially that of Schmidt with slight modifications due to the more restrictive nature of the problem. The condition on the g-function, that of finite first moment variational sum, fits nicely between the two more commonly used conditions, finite variation sum and exponentially decreasing variation.