Measure Theory, Metrology and Fuzzy Sets

Mathematical models of physical objects customarily rely on idealised Euclidean geometries, representing boundaries as crisp, continuous lines or surfaces. However, physical reality – ranging from the surface of a rubber ball to biological membranes – is characterised by microscopic roughness, textural undulations, and measurement ambiguity. This paper introduces a method to model such geometries using fuzzy valued functions. By understanding the radius of a parameterised curve outlining a physical object in polar or spherical coordinates not as a scalar, but as a fuzzy number derived from Gaussian measurement error models via truncated probability distributions, one captures both macroscopic shape and microscopic roughness. The paper further demonstrates how global geometric properties – Circumference, Area, Surface Area and Volume – can be derived as fuzzy numbers through the integration of these functions.