Measuring Asymptotic Convergence: A Unified Framework from Isotropic Infinity to Anisotropic Ends

Abstract: We develop a unified approach to defining a point at infinity for an arbitrary space and formalizing convergence to this point. Central to our work is a method to quantify and classify the rates at which functions approach their limits at infinity. Our framework applies to various settings (metric spaces, topological spaces, directed sets, measure spaces) by introducing an exhaustion of the space via an associated exhaustion function h. Using h, we adjoin an ideal point ωA to the space A and define convergence a \to ω_A in a manner intrinsic to A. To measure convergence rates, we introduce a family of parameterized norms, denoted ||f||{\infty,h,p}, which provides a refined classification of asymptotic behavior (e.g., distinguishing rates of order O(h^{-p})). Furthermore, the framework is extended to handle anisotropic spaces with multiple distinct ends by introducing a 'multi-exhaustion' formalism, allowing for a precise, directional analysis of convergence rates towards each asymptotic channel. This approach allows for a distinction between the global convergence captured by the norm and the purely asymptotic behavior at infinity, which can be analyzed via the limit superior of the convergence ratio. We further investigate the theoretical limits of this measure by establishing sufficient conditions (such as monotonicity) under which a finite norm guarantees convergence. The framework is shown to recover classical results, such as the Alexandroff one-point compactification and standard definitions of limits, while also providing a richer quantitative structure. Examples in each context are provided to illustrate the concepts.