upper function and lebesgue integrals

I am comparing theorem 10.6(c) and 10.14(a) in Apostol's Mathematical Analysis.

My question is this:

Are constant functions considered upper functions? They certainly seem to fit the definition 10.4 for upper functions:

A real-valued function f defined on an interval I is called an upper function on I, and we write http://www.physicsforums.com/latex_i.../1397581-0.png, if there exists an increasing sequence of step functions http://www.physicsforums.com/latex_i.../1397581-1.png such that:
http://www.physicsforums.com/latex_i.../1397581-2.png
and
http://www.physicsforums.com/latex_i.../1397581-3.png

There there's definition 10.12 for Lebesgue integrable functions:

We denote by L(I) the set of all functions f of the form f = u - v where http://www.physicsforums.com/latex_i.../1397581-4.png and http://www.physicsforums.com/latex_i.../1397581-5.png. Each function f in L(I) is said to be Lebesgue-integrable on I and its integral is defined by the equation http://www.physicsforums.com/latex_i.../1397581-6.png

Here's thm 10.14(a):

http://www.physicsforums.com/latex_i.../1397581-7.png

Assuming that a constant function is an upper function, let v = 0. Choose some arbitrary upper function u. Let f = u - 0 and thus f is a member of L(I). Then http://www.physicsforums.com/latex_i.../1397581-8.png. By thm 10.14, http://www.physicsforums.com/latex_i.../1397581-9.png. Thus http://www.physicsforums.com/latex_i...1397581-10.png

But it is not always true that

integral -u = - integral u

when u is an upper function according to Thm 10.6(b):

http://www.physicsforums.com/latex_i...1397581-11.png

Notice that c must be >= 0. There's even a problem in the text that shows that thm 10.6(b) isn't always true if that assumption is violated.

My point is that if constant functions are allowed as upper functions, then thm 10.14(a) implies thm 10.6(b) should be true for all real c, not just non-negative c. Any thoughts on where my logic is wrong?

I see my mistake. A constant is an upper function, but:

if u is a member of U(I) and f = u. Then f http://www.physicsforums.com/latex_i.../1398133-1.png. But http://www.physicsforums.com/latex_i.../1398133-2.png. The negative must be on the outside of the integral sign for u.