The establishment and philosophical implications of Hausdorff measure

Abstract

With the deepening of rigorous analysis in mathematics since the modern era, irregular pathological mathematical objects such as the Weierstrass function, Cantor set, and Koch curve, have been produced one after another, which result in some phenomena and problems that are difficult to explain by traditional mathematical theory. The measurement problem of the Cantor set is typical in these problems. According to the conventional mathematical theory, the measure of the Cantor set is zero. However, the zero-measure set constitutes infinite points which brings people confusion intuitively. Inspired by Fréchet’s fractional dimension, Hausdorff found the internal relationship between the fractional dimension and irregular pathological set based on Carathéodory measure and further created the Hausdorff measure, the predecessor of fractal measure, which not only solved the measurement problem of Cantor set but also paved the way of fractal theory. As the metric of nature, the fractal measure has unique and profound implications in ontology, epistemology, and methodology, in addition to accurately characterizing the magnitude of irregular objective things.