Abstract
This chapter is devoted to the basics in the integration theory, both in the Riemann and Lebesgue sense. Lebesgue’s theory is intertwined with the measure theory developed in Chapter 3. This allows for a finer analysis of functions and convergence.
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Notes
- 1.
Part of the standard calculus books efforts are devoted to the so-called “Calculus of Antiderivative —or ‘Primitive'— Functions” methods, some of them are included here in exercises collected at Sect. 13.5.2.
- 2.
The difference between the Riemann and the Lebesgue integration is sometimes described as follows: to count the total value of a pile of coins, the Riemann approach consists in taking coins one by one and adding their values, while the Lebesgue approach proceeds by grouping first the two-dollar coins, then the one-dollar ones, then the quarters, etc., and finally counting the number of coins in each of the piles.
- 3.
Since a step function s is defined (see Definition 721) disregarding the values at the end points of the finite family of subintervals that define s—and then s can be redefined to be, say, 0 at those points without affecting the integral—, we may omit the term (a.e.) here.
- 4.
See, however, Corollary 796.
- 5.
For the bounded-variation version see Corollary 796.
- 6.
For an example of how to use properly the Integration by Parts Theorem 705—formulated for the Riemann integral—in this context, see footnote 7.
- 7.
Theorem 705 was established for Riemann integrals. However, it can be used here in the following way: fix \({n\in\mathbb{N}}\) and consider the functions
$$f_n(x):=\left\{\begin{array}{@{}ll} x^s & \hbox{if} x\in I_n:=[1/n,n],\\ 0 & \hbox{if} x\in(0,+\infty)\setminus I_n,\end{array} \right.$$and
$$g_n(x):=\left\{\begin{array}{@{}ll} e^{-x} & \hbox{if} x\in I_n:=[1/n,n],\\ 0 & \hbox{if} x\in(0,+\infty)\setminus I_n,\end{array} \right.$$Both f n and g n are continuous on I n and continuously differentiable on \((1/n,n)\), hence
$$\begin{aligned} {\int_{(0,+\infty)}f_ng' _n=\int_{1/n}^{n}f_ng' _n}\\ &&=f_ng_n\Big|_{1/n}^{n}-\int_{1/n}^{n}f' _ng_n=n^se^{-n}-\left(\frac{1}{n}\right)^{s}e^{-1/n}-\int_{(0,+\infty)}f' _ng_n.\end{aligned}$$Observe that \(n^{s}e^{-n}\rightarrow 0\), \((1/n)^se^{-1/n}\rightarrow 0\) as \(n\rightarrow \infty\), and \(f_ng' _n\uparrow fg'\) and \(f' _ng_n\uparrow f' g\) pointwise on \((0,+\infty)\) as \(n\rightarrow \infty\). Since \(fg'\) and \(f' g\) are in \({\cal L}(0,+\infty)\), the use of the Dominated Convergence Theorem 750 proves the statement.
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Montesinos, V., Zizler, P., Zizler, V. (2015). Integration. In: An Introduction to Modern Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-12481-0_7
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DOI: https://doi.org/10.1007/978-3-319-12481-0_7
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