Help with Apostol's "Calculus, vol. 1", Section 1.18

[Born]

In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem 1.11, deals with the area of the graph of the function of theorem 1.10. (I have attached two excrepts from Apostol's book, one per theorem.)

I am having problems understansing Apostol's logic in theorem 1.11 where he states:

"The argument used to prove Theorem 1.10 also shows that Q’ is measurable and that a(Q’) = a(Q)."

I don see how he could argue this, being that $Q'=\{(x,y)|a \le x \le b, 0 \le y < f(x) \}$ which implies that $S \subseteq Q'$ is not true for all step regions S (since S may contain a point of the graph of $f(x)$ which Q', by definition, can't).

Thanks in advanced for any help.

 

[RUber]

I think the method is the same, but the definition of S would have to change if a point in S was equal to f(x).
So for the measure of Q', you would have sets S' and T', but the result would be the same--that the area of a line (or curve) is zero.

 

[STEMucator]

You can make the points $(x, y)$ in the set $Q'$ sufficiently close to the graph of $y = f(x)$. So the areas are indeed equal.

 

[Born]

RUber, I see what you're saying, however the new step regions (S' and T') would produce functions bearing the following relationship with the function$f(x)$: $s'(x) < f(x) \le t'(x)$. Which doesn't help since the definition of the integral requires "$\le$" for both step function inequalities.

Zondrina, The idea is making the of make the points equal since by definition of Q': $Q \neq Q'$. So no matter how close I get too the points of $f(x)$, I would not be able to put all points of both regions in a one-to-one correspondence in order to then argue by congruence of regions.

 

[RUber]

You can define a sequence s_n(x) such that the limit as n goes to infinity of s_n(x) = f(x). In this way, you will still satisfy the requirement that the intervals will converge on the true integral.

 

[Born]

I get what you're saying. The only problem is that Apostol still hasn't mentioned sequences. The only relevant thing I can think of that has been covered in the book till now is the Least Upper Bound Property of Numbers. Which would simply state that $f(x)$ is the supremum for $Q'$