91Ó°ÊÓ

Let \(\mathbf{I}\) be a generalized rectangle in \(\mathbb{R}^{n}\) and suppose that the function \(f: \mathbf{I} \rightarrow \mathbb{R}\) is integrable. Assume that \(f(\mathbf{x}) \geq 0\) if \(\mathbf{x}\) is a point in \(\mathbf{I}\) with a rational component. Prove that \(\int_{I} f \geq 0\).

Show that a subset \(S\) of \(\mathbb{R}^{n}\) that has Jordan content 0 has an empty interior. (Hint. By Exercise \(7, S\) cannot contain a generalized rectangle.

Let I be a generalized rectangle in \(\mathbb{R}^{2}\) and suppose that the bounded function \(f: \mathbf{I} \rightarrow \mathbb{R}\) has the value 0 on the interior of I. Show that \(f: \mathbf{I} \rightarrow \mathbb{R}\) is integrable and that \(\int_{\mathbf{I}} f=0 .\) Is the same result true for a generalized rectangle \(\mathbf{I}\) in \(\mathbb{R}^{n}\) ?

Let \(I\) be a generalized rectangle in \(\mathbb{R}^{n}\) and suppose that the function \(f: \mathbf{I} \rightarrow \mathbb{R}\) is integrable. Let the number \(M\) have the property that \(|f(\mathbf{x})| \leq M\) for all \(\mathbf{x}\) in \(\mathbf{I}\). Prove that $$ \left|\int_{\mathbf{I}} f\right| \leq M \cdot \operatorname{vol} \mathbf{I} $$