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Chapter 10: Problem 9

If \(f \in \mathcal{L}[a, b]\), show that \(f^{2}\) is not necessarily in \(\mathcal{L}[a, b]\).

Short Answer

Expert verified
The function \( f(x) = \frac{1}{\sqrt{x}} \) is a counter-example to prove that the square of an Lebesgue integrable function is not necessarily Lebesgue integrable. The function \( f \) is integrable over the interval [0, 1] as its integral is finite. However, the square of this function \( f^{2} \) is not integrable over the same interval as the integral of \( f^{2} \) diverges.

Step by step solution

01

Define a counter-example

Let's choose an function \( f \) as the counter-example for this problem. The function shall be defined as \( f(x) = \frac{1}{\sqrt{x}}\) on the interval \( (0, 1] \) and \( f(0) = 0 \) to ensure it's within the limit [a, b].
02

Show function is integrable

Prove the function is integrable. To do so, calculate the integral of \( f \) over the interval from 0 to 1. This is \( \int_{0}^{1} f(x) dx = \int_{0}^{1} \frac{1}{\sqrt{x}} dx = 2 \sqrt{x} \Big|_0^1 = 2(1-0) = 2 \) . Here, calculation shows that the function is integrable because the integral is finite.
03

Show square of function is not integrable

Calculate the Lebesgue integral of the square of \( f \) over the same interval. This is \( \int_{0}^{1} f^{2}(x) dx = \int_{0}^{1} \frac{1}{x} dx \). As this integral diverges, \( f^{2} \) is not Lebesgue integrable on the interval \( [0, 1] \). Therefore, \( f^{2} \) is not necessarily in \( \mathcal{L}[0, 1] \), even when f is Lebesgue integrable.

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Most popular questions from this chapter

Let \(f(x):=1 / x\) for \(x \in(0,1]\) and \(f(0):=0 ;\) then \(f\) is continuous except at \(x=0 .\) Show that \(f\) does not belong to \(\mathcal{R}^{*}[0,1] \cdot\) [Hint: Compare \(f\) with \(s_{n}(x):=1\) on \((1 / 2,1], s_{n}(x):=2\) on \((1 / 3,1 / 2], s_{n}(x):=3\) on \((1 / 4,1 / 3], \cdots, s_{n}(x):=n\) on \(\left.[0,1 / n] .\right]\)

Let \(\delta\) be a gauge on \([a, b]\) and let \(\dot{\mathcal{P}}=\left\\{\left(\left[x_{i-1}, x_{i}\right], t_{i}\right)\right\\}_{i=1}^{n}\) be a \(\delta\) -fine partition of \([a, b]\). (a) Show that \(00\), and if \(\mathcal{Q}\) is a tagged partition of \([a, b]\) such that we have \(\|\dot{\mathcal{Q}}\| \leq \delta_{*}\), show that \(\mathcal{Q}\) is \(\delta\) -fine. (d) If \(\varepsilon=1\), show that the gauge \(\delta_{1}\) in Example \(10.1 .4(a)\) has the property that \(\inf \left\\{\delta_{1}(t): t \in\right.\) \([0,1]\\}=0\)

Show that the integral \(\int_{0}^{\infty}(1 / \sqrt{x}) \sin x d x\) converges. [Hint: Integrate by Parts.]

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