\input vanilla.sty \magnification=1200 \baselineskip=20pt \nopagenumbers \centerline{\bf Preliminary Examation} \centerline{\bf Real Analysis} \centerline{\bf January 1998} \bigskip \noindent Do all of problems 1-2, and as many of the remaining problems as possible. \bigskip \item{1.} \underbar{State the following theorems}. (a) Monotone Convergence Theorem (b) Fatou's Lemma (c) Dominated Convergence Theorem (d) H\"older's Inequality (e) Fubini's Theorem (f) Egoroff's Theorem \bigskip \item{2.} Prove 1(a), (b), and (c) \underbar{in that order}. \bigskip \item{3.} Let $\lambda$ be Lebesgue measure and $\mu$ be counting measure both regarded as Borel measures on $I=[0,1]$. Let $\Delta$ be the diagonal in $I\times I$; $\Delta=\{(x,y)|x=y\}$. \itemitem{(a)} Show that $\Delta$ is measurable (with respect to the product measure on Borel subsets of $I\times I$. \itemitem{(b)} Let $f$ be the characteristic function in $\Delta$. Compute the integrals: $\dsize\int_I(\int_1 fd\lambda)d\mu$, $\dsize\int_I(\int_I f(d\mu)d\lambda)$ and $\dsize\int_{I\times I} fd\mu \times d\lambda$. \itemitem{(c)} Reconcile with Fubini's Theorem. \bigskip \item{4.} Let $m$ be Lebesgue measure on $X=[0,1]$, and let $\mu$ be a measure on the Lebesgue sets with $\mu(X)=1$, and $\mu \sim m$ (i.e. $\mu$ and $m$ have the same sets of measure zero). Prove there exists a measurable set $A$ such that $\mu(A)=1/2$. \bigskip \item{5.} Prove that the product of two measurable functions is a measurable function. \bigskip \item{6.} (a) Define bounded variation of a function $f$ on $[a,b]$. \itemitem{(b)} Show that if $f$ is a function of bounded variation on $[a,b]$, then $f$ has at most a countable number of discontinuities. \bigskip \item{7.} Let $f_n$ be a sequence of measurable functions such that $f_n(x)\to f(x)$ almost everywhere, and suppose that $\sup \dsize\int^1_0 |f_n(x)|dx <\infty$. \medskip (a) Show that $f$ is measurable and that $\dsize\int^1_0 |f(x)|dx <\infty$. \medskip (b) Does $\dsize\lim_{n\to \infty} \int^1_0 f_n(x)dx = \int^1_0 f(x)dx$? \bigskip \item{8.} Give an example of a sequence $f_n$ of \underbar{continuous} functions on [0,1] converging pointwise to a \underbar{continuous} function $f$ on [0,1] such that $\dsize\int f_n \notin \int f$. \bigskip \item{9.} Define a sequence of measures $(\mu_n)$ on the Lebesgue measurable subsets of $[0,1]$ by $\mu_n(A) = \dsize\int^1_0 I_A(x) nx^{n-1} dx$, $n=1,2,3,\dots$ \ . \medskip (a) Verify that $\dsize\lim_{n\to \infty} \mu_n([a,b]) = 0$ if $0