Area will be defined as a function satisfying certain conditions (20.3). The so-called Lebesgue measure gives an example of such function. In particular a construction of Lebesgue measure implies the existence of area function. This construction is included in any textbook in real analysis.
Based solely on its existence, we develop the concept of area with no cheating. We choose this approach since any rigorous introduction to area is tedious. We do not want to cheat and at the same time we do not want to waste your time; soon or later you will have to learn Lebesgue measure if it is not done already.