rel="nofollow" href="#fb3_img_img_424cabda-8b2d-5f19-b675-c9e0ee50e747.png" alt="upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis subset-of upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis period"/> In order to prove that for every
As we already mentioned, we will consider a more general definition of the Kurzweil integral in Chapter 2. Thus, in the remaining of this chapter, we refer to the integrals
as Perron–Stieltjes integrals, where
As it should be expected, the above integrals are linear and additive over nonoverlapping intervals. These facts will be put aside for a while, because in Chapter 2 they will be proved for the more general form of the Kurzweil integral. In the meantime, we present a simple example of a function which is Riemann improper integrable (and, hence, also Perron integrable, due to Theorem 2.9), but it is not Lebesgue integrable (because it is not absolutely integrable).
Example 1.43: Let
Another example is also needed at this point. Borrowed from [73, example 2.1], the example below exhibits a function
