91Ó°ÊÓ

Suppose \((X, \mathcal{S}, \mu)\) and \((Y, \mathcal{T}, v)\) are \(\sigma\) -finite measure spaces. Prove that if \(\omega\) is a measure on \(\mathcal{S} \otimes \mathcal{T}\) such that \(\omega(A \times B)=\mu(A) v(B)\) for all \(A \in \mathcal{S}\) and all \(B \in \mathcal{T},\) then \(\omega=\mu \times v\) [The exercise above means that \(\mu \times v\) is the unique measure on \(\mathcal{S} \otimes \mathcal{T}\) that behaves as we expect on measurable rectangles.]

Suppose \(F_{1}\) is a nonempty subset of \(\mathbf{R}^{m}\) and \(F_{2}\) is a nonempty subset of \(\mathbf{R}^{n}\). Prove that \(F_{1} \times F_{2}\) is a closed subset of \(\mathbf{R}^{m} \times \mathbf{R}^{n}\) if and only if \(F_{1}\) is a closed subset of \(\mathbf{R}^{m}\) and \(F_{2}\) is a closed subset of \(\mathbf{R}^{n}\)

Suppose \(E\) is a subset of \(\mathbf{R}^{m} \times \mathbf{R}^{n}\) and $$ A=\left\\{x \in \mathbf{R}^{m}:(x, y) \in E \text { for some } y \in \mathbf{R}^{n}\right\\} $$ (a) Prove that if \(E\) is an open subset of \(\mathbf{R}^{m} \times \mathbf{R}^{n},\) then \(A\) is an open subset of \(\mathbf{R}^{m}\) (b) Prove or give a counterexample: If \(E\) is a closed subset of \(\mathbf{R}^{m} \times \mathbf{R}^{n},\) then \(A\) is a closed subset of \(\mathbf{R}^{m}\).

For readers familiar with the gamma function \(\Gamma:\) Prove that $$ \lambda_{n}\left(\mathbf{B}_{n}\right)=\frac{\pi^{n / 2}}{\Gamma\left(\frac{n}{2}+1\right)} $$ for every positive integer \(n\).

Define \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) by $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0), \\ 0 & \text { if }(x, y)=(0,0) . \end{array}\right. $$ (a) Prove that \(D_{1}\left(D_{2} f\right)\) and \(D_{2}\left(D_{1} f\right)\) exist everywhere on \(\mathbf{R}^{2}\). (b) Show that \(\left(D_{1}\left(D_{2} f\right)\right)(0,0) \neq\left(D_{2}\left(D_{1} f\right)\right)(0,0)\). (c) Explain why (b) does not violate 5.48 .