Able to define definable

@Agnishom Chattopadhyay – You should read up on measure theory and real analysis then.

In classical Euclidean geometry, length is a fundamental property of space. However, by moving from Euclidean geometry to analytic geometry in $\mathbb{R}^{n}$ , you no longer have length as a fundamental concept. For example, $|(0,1)| = |(0,2)| = 2^{\aleph_0}$ ; however, they clearly have different "lengths". Thus, the Lebesgue measure of a set arises as a way to define length, area, volume, etc in $\mathbb{R}^{n}$ . It finds use in being the main ingredient in creating the Lebesgue integral, a more powerful tool than the regular Riemann sum.