Area and Integral of step functions in Tom Apostol's Calculus vol. $1$

Question

At the beginning of the section $1.18$ of Tom Apostol's Calculus vol $1.$ (second edition), it is written that from the area properties introduced in section $1.6$, he proved that the area of the ordinate set of a nonnegative step function is equal to the integral of the function.

However, in the section $1.12$, where he is defining the integral for step functions, he just says that the definition is made such that the integral is equal to the area of the function's ordinate set, but I do not see any proof for that later.

Is there a proof maybe in an earlier version of the book, and it was removed from the second edition (which is the one I have)? In particular, I would be interested to see the proof that the step function integral satisfies the exhaustion property (i.e. Axiom $6$ of an area function).

Thank you!

Answer 1

I still didn't find the proof in the book, but it seems it could be derived from the Additive and the Choice of scale properties, which are the axioms of an area function. Namely:

1. Any rectangle is measurable and has area $a(R) = hk$ (by Property 5).
2. A set of rectangles is also measurable and has the area which is the sum of the areas of the individual rectangles (by Property 2).

The definition of the integral of a non-negative step function is the sum of the areas of individual rectangles, which fits in the Property 2. Therefore, by being the part of the assumed area function, the integral of the non-negative step functions also satisfies the other properties too, including the exhaustion property.

Please let me know if I went wrong somewhere.