Generalized Inverses of Increasing Functions and Lebesgue Decomposition

The reader should be aware of the explanatory nature of this article. Its main goal is to introduce to a broader vision of a topic than a more focused research paper, demonstrating some new results but mainly starting from some general consideration to build an overview of a theme with links to connected problems. Our original question was related to the height of random growing trees. When investigating limit processes, we may consider some measures that are defined by increasing functions and their generalized inverses. And this leads to the analysis of Lebesgue decomposition of generalized inverses. Moreover, since the measures that motivated us initially are stochastic, there arises the idea of studying the continuity property of this transform in order to take limits. When scaling growing processes like trees, time origin and scale can be replaced by another process. This leads us to a clock metaphor, to consider an increasing function as a clock reading from a given timeline. This is nothing more than an explanatory vision, not a mathematical concept, but this is the nature of this paper. So we consider an increasing function as a time change between two timelines; it leads to the idea that an increasing function and its generalized inverse play symmetric roles. We then introduce a universal time that links symmetrically an increasing function and its generalized inverse. We show how both are smoothly defined from this universal time. This allows to describe the Lebesgue decomposition for both an increasing function and its generalized inverse.