Lebesgue measure

Lebesgue measure is a mathematical way to assign a size or length to a set of points in space, extending our usual concept of length, area, or volume. Unlike traditional methods that work well for simple shapes, Lebesgue measure can handle very irregular sets, like fractals, by focusing on how they can be approximated from simple shapes. It ensures that the measure aligns with our intuition for standard shapes and provides a consistent method to measure more complex or "thin" sets that traditional approaches might find problematic.