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Chapter 10: Problem 11

(a) Give an example of a function \(f \in \mathcal{R}^{*}[0,1]\) such that max \(\\{f, 0\\}\) does not belong to \(\mathcal{R}^{*}[0,1]\). (b) Can you give an example of \(f \in \mathcal{L}[0,1]\) such that \(\max \\{f, 0\\} \notin \mathcal{L}[0,1] ?\)

Short Answer

Expert verified
The indicator function of the rationals, defined as \(f = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases}\), is an example of a function for both part (a) and (b) of this exercise.

Step by step solution

01

Find function for (a)

For problem (a), we need a function which is Riemann-integrable but where the maximum of the function and 0 does not have a limit. An example of such a function could be the indicator function of the rationals, denoted by \(1_{\mathbb{Q}}\), defined as: \[f = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases}\] However, \(\max\{f, 0\}\) is equal to \(f\) itself which does not belong to \(\mathcal{R}^{*}[0,1]\) because it does not have a limit at any point in \([0,1]\) and therefore is not Riemann-integrable.
02

Find function for (b)

For problem (b), we need a function which is Lebesgue-integrable but where the maximum of that function and 0 is not in \(\mathcal{L}[0,1]\). In Lebesgue integration theory, every Riemann-integrable function is also Lebesgue-integrable. That is, if \(f \in \mathcal{R}^{*}[0,1]\) then \(f \in \(\mathcal{L}[0,1]\). This in turn implies that if \(\max\{f, 0\} \notin \(\mathcal{R}^{*}[0,1]\), then \(\max\{f, 0\} \notin \(\mathcal{L}[0,1]\). Therefore, an example of such a function for this problem would be the same function as described in the first problem, namely the indicator function of the rationals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lebesgue Integration
Lebesgue integration is a powerful and flexible method of integration that extends the concept of integrating functions. Unlike Riemann integration, which partitions the x-axis, Lebesgue integration partitions the y-axis. This allows it to handle a broader class of functions.
For example, consider functions that are not easy to integrate using the traditional Riemann method, like the indicator function over the rational numbers. Lebesgue integration works well here because it focuses on how the function's range is distributed. It involves measuring the size of the set where the function takes certain values.
This characteristic makes Lebesgue integration useful in both practical and theoretical applications, such as probability theory and real analysis. It broadens the types of functions that can be integrated, including functions with many discontinuities. Typically, all Riemann-integrable functions are also Lebesgue-integrable, but the reverse isn't always true.
Indicator Function
An indicator function is a simple mathematical tool used to indicate the presence of an element within a certain set. This function is defined to be 1 for elements in the set and 0 for elements outside the set.
In mathematical notation, the indicator function for a set \( S \) is written as \( 1_S(x) \). It is defined as follows:
  • \( 1 \) if \( x \) belongs to \( S \)
  • \( 0 \) if \( x \) does not belong to \( S \)
Indicator functions are particularly useful in simplifying complex expressions and are widely used in probability and statistics.
One interesting example is the indicator function for rational numbers within a specified interval, such as \([0, 1]\). This is used in the exercise to illustrate how a function can be Riemann-integrable or not, based on its behavior and the definition used.
Rational Numbers
Rational numbers are numbers that can be expressed as the quotient of two integers, where the numerator is an integer and the denominator is a non-zero integer. In essence, rational numbers can be written in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \).
They include numbers like \( \frac{1}{2}, -3, \frac{22}{7}, 5.0 \) and any other fraction or whole number.
Understanding rational numbers is crucial for recognizing their properties and how they relate to functions and sets. The set of rational numbers \( \mathbb{Q} \) is dense in the real numbers, meaning in any interval of real numbers, there are infinitely many rationals.
When considering functions over an interval, differentiating between rational numbers and their complements (irrational numbers) can illustrate certain integration properties, as seen with the indicator function over rational numbers used in the exercise.

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Most popular questions from this chapter

Let \(g_{n}(x):=-1\) for \(x \in[-1,-1 / n)\), let \(g_{n}(x):=n x\) for \(x \in[-1 / n, 1 / n]\) and let \(g_{n}(x):=1\) for \(x \in\) \((1 / n, 1]\). Show that \(\| g_{m}-g_{n} \mid \rightarrow 0\) as \(m, n \rightarrow \infty\), so that the Completeness Theorem 10.2.12 implies that there exists \(g \in \mathcal{L}[-1,1]\) such that \(\left(g_{n}\right)\) converges to \(g\) in \(\mathcal{L}[-1,1] .\) Find such a function \(g\).

A set \(E\) in \([a, b]\) is said to be (Lebesgue) measurable if its characteristic function \(\mathbf{1}_{E}\) (defined by \(\mathbf{1}_{E}(x):=1\) if \(x \in E\) and \(\mathbf{1}_{E}(x):=0\) if \(x \in[a, b] \backslash E\) ) belongs to \(\mathcal{M}[a, b] .\) We will denote the collection of measurable sets in \([a, b]\) by \(\mathbb{M}[a, b] .\) In this exercise, we develop a number of properties of \(\mathbb{M}[a, b]\) (a) Show that \(E \in \mathbb{M}[a, b]\) if and only if \(\mathbf{1}_{E}\) belongs to \(\mathcal{R}^{*}[a, b]\). (b) Show that \(\emptyset \in \mathbb{M}[a, b]\) and that if \([c, d] \subseteq[a, b]\), then the intervals \([c, d],[c, d),(c, d]\), and \((c, d)\) are in \(\mathbb{M}[a, b]\) (c) Show that \(E \in \mathbb{M}[a, b]\) if and only if \(E^{\prime}:=[a, b] \backslash E\) is in \(\mathbb{M}[a, b]\). (d) If \(E\) and \(F\) are in \(\mathbb{M}[a, b]\), then \(E \cup F, E \cap F\) and \(E \backslash F\) are also in \(\mathbb{M}[a, b] .\) [Hint: Show that \(\mathbf{1}_{\mathrm{EUF}}=\max \left\\{\mathbf{1}_{E}, \mathbf{1}_{F}\right\\}\), etc. \(]\) (e) If \(\left(E_{k}\right)\) is an increasing sequence in \(\mathbb{M}[a, b]\), show that \(E:=\cup_{k=1}^{\infty} E_{k}\) is in \(\mathbb{M}[a, b] .\) Also, if \(\left(F_{k}\right)\) is a decreasing sequence in \(\mathbb{M}[a, b]\), show that \(F:=\cap_{k=1}^{\infty} F_{k}\) is in \(M[a, b] .\) [Hint: Apply Theorem \(10.4 .9(\mathrm{c}) .]\) (f) If \(\left(E_{k}\right)\) is any sequence in \(\mathbb{M}[a, b]\), show that \(\cup_{k=1}^{\infty} E_{k}\) and \(\cap_{k=1}^{\infty} E_{k}\) are in \(\mathbb{M}[a, b]\).

Let \(M(x):=\ln |x|\) for \(x \neq 0\) and \(M(0):=0\). Show that \(M^{\prime}(x)=1 / x\) for all \(x \neq 0\). Explain why it does not follow that \(\int_{-2}^{2}(1 / x) d x=\ln |-2|-\ln 2=0\)

If \(f:[a, b] \rightarrow \mathbb{R}\) is continuous and if \(p \in \mathcal{R}^{*}[a, b]\) does not change sign on \([a, b]\), and if \(f p \in \mathcal{R}^{*}[a, b]\), then there exists \(\xi \in[a, b]\) such that \(\int_{a}^{b} f p=f(\xi) \int_{a}^{b} p\). (This is a generalization of Exercise 7.2.16; it is called the First Mean Value Theorem for integrals.)

. (a) Show that the integral \(\int_{1}^{\infty} x^{-1} \ln x d x\) does not converge. (b) Show that if \(\alpha>1\), then \(\int_{1}^{\infty} x^{-\alpha} \ln x d x=1 /(\alpha-1)^{2}\).
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