I was studying Apostol's Calculus second edition (volume 1) chapter THE CONCEPTS OF INTEGRAL CALCULUS, section 1.6 The concept of area as a set function where I ran into this:
AXIOMATIC DEFINITION OF AREA. We assume there exists a class $M$ of measurable sets in the plane and a set function $a$, whose domain is $M$ , with the following properties: ... (page 58)
I don't know the definition of "measurable sets in the plane".