Inductive and projective limits of Banach spaces of measurable functions with order unites with respect to power parameter

We prove that a measurable function $f$ is bounded and invertible if and only if there exist at least two equivalent norms by order unit spaces with order units $f^\alpha$ and $f^\beta$ with $\alpha>\beta>0$. We show that it is natural to understand the limit of ordered vector spaces with order units $f^\alpha$ ($\alpha$ approaches to infinity) as a direct sum of one inductive and one projective limits. We also obtain some properties for the corresponding limit topologies.