Chapter 3

3-2. Let \(f: A \rightarrow \mathbf{R}\) be integrable and let \(g=f\) except at finitely many points. Show that \(g\) is integrable and \(\int_{A} f=\int_{A} g\).

3-3. Let \(f, g: A \rightarrow \mathbf{R}\) be integrable. (a) For any partition \(P\) of \(A\) and subrectangle \(S\), show that $$ \begin{aligned} m_{S}(f)+m_{S}(g) \leq m_{S}(f+g) \quad \text { and } \quad M_{S}(f+g) \\ & \leq M_{s}(f)+M_{S}(g) \end{aligned} $$ and therefore $$ \begin{aligned} L(f, P)+L(g, P) \leq L(f+g, P) \quad \text { and } \quad & U(f+\theta, P) \\ & \leq U(f, P)+U(g, P) \end{aligned} $$ (b) Show that \(f+g\) is integrable and \(\int_{A} f+g=\int_{A} f+\int_{A} g\). (c) For any constant \(c\), show that \(\int_{A} c f=c \int_{A} f\).

3-5. Let \(f, g: A \rightarrow \mathbf{R}\) be integrable and suppose \(f \leq g\). Show that \(\int_{A} f \leq \int_{A} g\)

3-6. If \(f: A \rightarrow \mathbf{R}\) is integrable, show that \(|f|\) is integrable and \(\left|\int_{A} f\right| \leq\) \(\int_{A}|f| .\)

3-7. Let \(f:[0,1] \times[0,1] \rightarrow \mathbf{R}\) be defined by $$ f(x, y)= \begin{cases}0 & x \text { irrational } \\ 0 & x \text { rational, } y \text { irrational, } \\ 1 / q & x \text { rational, } y=p / q \text { in lowest terms. }\end{cases} $$ Show that \(f\) is integrable and \(\int_{[0,1] \times[0,1]} f=0\)

\(3-10 .\) (a) If \(C^{\prime}\) is a set of content 0, show that the boundary of \(C\) has content 0 . (b) Give an example of a bounded set \(C\) of measure 0 such that the boundary of \(C\) does not have measure \(0 .\)

3 -13. \(^{*}\) (a) Show that the collection of all rectangles \(\left[a_{1}, b_{1}\right] \times \times \times\) \(\left[a_{n}, b_{n}\right]\) with all \(a_{i}\) and \(b_{i}\) rational can be arranged in a sequence. (b) If \(A \subset \mathbf{R}^{n}\) is any set and \(\mathcal{O}\) is an open cover of \(A\), show that there is a sequence \(U_{1}, U_{2}, U_{3}, \ldots\) of members of \(O\) which also cover \(A\). Hint: For each \(x \in A\) there is a rectangle \(B=\left[a_{1}, b_{1}\right] \times\) \(\therefore \times\left[a_{n}, b_{n}\right]\) with all \(a_{i}\) and \(b_{i}\) rational such that \(x \in B \subset U\) for some \(U \in \mathcal{O}\)

3-14. Show that if \(\therefore \cdot g: A \rightarrow \mathbf{R}\) are integrable, so is \(f \cdot g .\)

3-18. If \(f: A \rightarrow \mathbf{R}\) is non-negative and \(\int_{A} f=0\), show that \(\\{x: f(x) \neq 0\\}\) has measure 0. Hint: Prove that \(\\{x: f(x)>1 / n\\}\) has content 0 .

3-23. Let \(C \subset A \times B\) be a set of content \(0 .\) Let \(A^{\prime} \subset A\) be the set of all \(x \in A\) such that \(\\{y \in B:(x, y) \in C\\}\) is not of content \(0 .\) Show that \(A^{\prime}\) is a set of measure 0. Hint \(: \chi C\) is integrable and \(\int_{A \times B \chi C}=\int_{A} \mathfrak{U}=\int_{A} \mathcal{L}\), so \(\int_{A} \mathfrak{U}-\mathcal{L}=0\)

3-25. Use induction on \(n\) to show that \(\left[a_{1}, b_{1}\right] \times \times\left[a_{n}, b_{n}\right]\) is not a set of measure 0 (or content 0\()\) if \(a_{i}

3-26. Let \(f:[a, b] \rightarrow \mathbf{R}\) be integrable and non-negative and let \(A_{f}=\) \(\\{(x, y): a \leq x \leq b\) and \(0 \leq y \leq f(x)\\} .\) Show that \(A_{f}\) is Jordanmeasurable and has area \(\int_{a}^{b} f\).

3-27. If \(f:[a, b] \times[a, b] \rightarrow \mathbf{R}\) is continuous, show that $$ \int_{a}^{b} \int_{a}^{y} f(x, y) d x d y=\int_{a} \int_{x} f(x, y) d y d x $$ Hint: Compute \(\int c f\) in two different ways for a suitable set \(C \subset[a, b] \times[a, b]\)

3-31. If \(A=\left[a_{1}, b_{1}\right] \times \times\left[a_{n}, b_{n}\right]\) and \(f: A \rightarrow \mathbf{R}\) is continuous, define \(F: A \rightarrow \mathbf{R}\) by $$ F(x)=\int_{\left[a_{1}, x^{1}\right] \times \cdots \times\left[a_{n}, x^{n}\right]} f $$ What is \(D_{i} F(x)\), for \(x\) in the interior of \(A ?\)

3-32. \({ }^{*}\) Let \(f:[a, b] \times[c, d] \rightarrow \mathbf{R}\) be continuous and suppose \(D_{2} f\) is continuous. Define \(F(y)=\int_{a}^{b} f(x, y) d x .\) Prove Leibnitz's rule: \(F^{\prime}(y)\) \(=\int_{a}^{b} D_{2} f(x, y) d x . \quad\) Hint \(: F(y)=\int_{a}^{b} f(x, y) d x=\int_{a}^{b}\left(\int_{c}^{v} D_{2} f(x, y) d y+\right.\) \(f(x, c)) d x\). (The proof will show that continuity of \(D_{2} f\) may be replaced by considerably weaker hypotheses.)

3-33. If \(f:[a, b] \times[c, d] \rightarrow \mathbf{R}\) is continuous and \(D_{2} f\) is continuous, define \(F(x, y)=\int_{a}^{x} f(t, y) d t\) (a) Find \(D_{1} F\) and \(D_{2} F\). (b) If \(G(x)=\int_{a}^{\theta(x)} f(t, x) d t\), find \(G^{\prime}(x)\).

3-35. \(^{*}(a)\) Let \(g: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}\) be a linear transformation of one of the following types: $$ \begin{aligned} &\left\\{\begin{array}{l} g\left(e_{i}\right)=e_{i} \\ g\left(e_{j}\right)=a e_{j} \end{array} \quad i \neq j\right. \\ &\left\\{\begin{array}{l} g\left(e_{i}\right)=e_{i} \quad i \neq j \\ g\left(e_{j}\right)=e_{j}+e_{k} \end{array}\right. \\ &\left\\{\begin{array}{l} g\left(e_{k}\right)=e_{k} \quad k \neq i, j \\ g\left(e_{i}\right)=e_{j} \\ g\left(e_{j}\right)=e_{i} \end{array}\right. \end{aligned} $$ If \(U\) is a rectangle, show that the volume of \(g(U)\) is \(|\operatorname{det} g| \cdot v(U)\). (b) Prove that \(|\operatorname{det} g| \cdot v(U)\) is the volume of \(g(U)\) for any linear transformation \(g: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}, \quad\) Hint \(:\) If det \(g \neq 0\), then \(g\) is the composition of linear transformations of the type considered in (a).

3-36. (Cavalieri's principle). Tet \(A\) and \(B\) be Jordan-measurable subsets of \(\mathbf{R}^{3} . \quad\) Let \(A_{c}=\\{(x, y):(x, y, c) \in A\\}\) and define \(B_{c}\) similarly. Suppose each \(A_{c}\) and \(B_{c}\) are Jordan-measurable and have the same area. Show that \(A\) and \(B\) have the same volume.