Chapter 6

Let \(D\) be a bounded subset of \(\mathbb{R}\) and \(f: D \rightarrow \mathbb{R}\) be a bounded function. If the boundary \(\partial D\) of \(D\) is of content zero and if the set of discontinuities of \(f\) is also of content zero, then show that \(f\) is integrable. In particular, if \(D\) is of content zero, then show that \(f\) is integrable and its Riemann integral is equal to zero. (Compare Remark 6.8.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function. If the set of discontinuities of \(f\) is of content zero, show that \(f\) is integrable. Is the converse true? (Hint: Exercise 34 of Chapter 3 and Example 6.16.)

Let \(D\) be a bounded subset of \(\mathbb{R}\). Let \(1_{D}: D \rightarrow \mathbb{R}\) be defined by \(1_{D}(x):=1\) for all \(x \in D\). Prove the following statements: (i) \(1_{D}\) is integrable if and only if \(\partial D\) is of content zero. [Note: If \(1_{D}\) is integrable, then \(\int_{D} 1_{D}(x) d x\) is called the length of the set \(D .]\) (ii) The length of \(D\) is zero if and only if \(D\) is of content zero. (iii) If \(f: D \rightarrow \mathbb{R}\) is a bounded function and \(D_{0} \subseteq D\) is such that \(\partial D\) is of content zero, then \(f\) is integrable on \(D_{0}\).

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable, and \(g:[a, b] \rightarrow \mathbb{R}\) be a bounded function such that the set \(\\{x \in[a, b]: g(x) \neq f(x)\\}\) is of content zero. Show that \(g\) is integrable and $$ \int_{a}^{b} g(x) d x=\int_{a}^{b} f(x) d x $$ (Compare Proposition 6.12.)