Chapter 3

Let \(\nu\) be a signed measure on \((X, \mathcal{M})\). a. \(L^{1}(\nu)=L^{1}(|\nu|)\). b. If \(f \in L^{1}(\nu),\left|\int f d \nu\right| \leq \int|f| d|\nu|\). c. If \(E \in \mathcal{M},|\nu|(E)=\sup \left\\{\left|\int_{E} f d \nu\right|:|f| \leq 1\right\\}\).

Suppose \(\left\\{\nu_{j}\right\\}\) is a sequence of positive measures. If \(\nu_{j} \perp \mu\) for all \(j\), then \(\sum_{1}^{\infty} \nu_{j} \perp \mu ;\) and if \(\nu_{j} \ll \mu\) for all \(j\), then \(\sum_{1}^{\infty} \nu_{j} \ll \mu\).

Let \(\mu\) be a positive measure. A collection of functions \(\left\\{f_{\alpha}\right\\}_{\alpha \in A} \subset L^{1}(\mu)\) is called uniformly integrable if for every \(\epsilon>0\) there exists \(\delta>0\) such that \(\left|\int_{E} f_{\alpha} d \mu\right|<\epsilon\) for all \(\alpha \in A\) whenever \(\mu(E)<\delta .\) a. Any finite subset of \(L^{1}(\mu)\) is uniformly integrable. b. If \(\left\\{f_{n}\right\\}\) is a sequence in \(L^{1}(\mu)\) that converges in the \(L^{1}\) metric to \(f \in L^{1}(\mu)\). then \(\left\\{f_{n}\right\\}\) is uniformly integrable.

If \(\nu\) is an arbitrary signed measure and \(\mu\) is a \(\sigma\)-finite measure on \((X, \mathcal{M})\) such that \(\nu \ll \mu\), there exists an extended \(\mu\)-integrable function \(f: X \rightarrow[-\infty, \infty]\) such that \(d \nu=f d \mu\). Hints: a. It suffices to assume that \(\mu\) is finite and \(\nu\) is positive. b. With these assumptions, there exists \(E \in \mathcal{M}\) that is \(\sigma\)-finite for \(\nu\) such that \(\mu(E) \geq \mu(F)\) for all sets \(F\) that are \(\sigma\)-finite for \(\nu\). c. The Radon-Nikodym theorem applies on \(E\). If \(F \cap E=\mathscr{\text { , then either }}\) \(\nu(F)=\mu(F)=0\) or \(\mu(F)>0\) and \(|\nu(F)|=\infty\).

If \(E\) is a Borel set in \(\mathbb{R}^{n}\), the density \(D_{E}(x)\) of \(E\) at \(x\) is defined as $$ D_{E}(x)=\lim _{r \rightarrow 0} \frac{m(E \cap B(r, x))}{m(B(r, x))} $$ whenever the limit exists. a. Show that \(D_{E}(x)=1\) for a.e. \(x \in E\) and \(D_{E}(x)=0\) for a.e. \(x \in E^{c}\). b. Find examples of \(E\) and \(x\) such that \(D_{E}(x)\) is a given number \(\alpha \in(0,1)\), or such that \(D_{E}(x)\) does not exist.

If \(\lambda\) and \(\mu\) are positive, mutually singular Borel measures on \(\mathbb{R}^{n}\) and \(\lambda+\mu\) is regular, then so are \(\lambda\) and \(\mu\).

Construct an increasing function on \(\mathbb{R}\) whose set of discontinuities is \(\mathbb{Q}\).

Let \(F(x)=x^{2} \sin \left(x^{-1}\right)\) and \(G(x)=x^{2} \sin \left(x^{-2}\right)\) for \(x \neq 0\), and \(F(0)=\) \(G(0)=0\). a. \(F\) and \(G\) are differentiable everywhere (including \(x=0\) ). b. \(F \in B V([-1,1])\), but \(G \notin B V([-1,1])\).

If \(F\) and \(G\) are absolutely continuous on \([a, b]\), then so is \(F G\), and $$ \int_{a}^{b}\left(F G^{\prime}+G F^{\prime}\right)(x) d x=F(b) G(b)-F(a) G(a) $$

If \(f:[a, b] \rightarrow \mathbb{R}\), consider the graph of \(f\) as a subset of \(\mathrm{C}\), namely, \(\\{t+i f(t)\) : \(t \in[a, b]\\}\). The length \(L\) of this graph is by definition the supremum of the lengths of all inscribed polygons. (An "inscribed polygon" is the union of the line segments joining \(t_{j-1}+i f\left(t_{j-1}\right)\) to \(t_{j}+i f\left(t_{j}\right), 1 \leq j \leq n\), where \(\left.a=t_{0}<\cdots