Reading time: about 12 minutes (2592 words).

1998 April 03

Instructions. Do at least four problems in Part A, and at least two problems in Part B.


  1. Let $\{x_n\}_{n=1}^\infty$ be a bounded sequence of real numbers, and for each positive $n$ define $\hat{x}_n = sup _{k \geq n} x_k$.
    a. Explain why the limit $\ell = \lim_n \hat{x}_n$ exists.
    b. Prove that, for any $\epsilon > 0$ and positive integer $N$, there exists an integer $k$ such that $k\geq N$ and $|x_k - \ell| < \epsilon$.

  2. Let $C$ be a collection of subsets of the real line $\mathbb R$, and define $$A_\sigma (C) = \bigcap \{A: C \subset A \text{ and } A \text{ is a $\sigma$-algebra of subsets of } R\}.$$ a. Prove that $A_\sigma (C)$ is a $\sigma$-algebra, that $C \subset A_\sigma (C)$, and that $A_\sigma (C) \subset A$ for any other $\sigma$-algebra $A$ containing all the sets of $C$.
    b. Let $O$ be the collection of all finite open intervals in $\mathbb R$, and $F$ the collection of all finite closed intervals in $\mathbb R$. Show that $A_\sigma (O) = A_\sigma (F)$.

  3. Let $(X, A, \mu)$ be a measure space, and suppose $X = \bigcup_n X_n$, where $\{X_n\}_{n=1}^{\infty}$ is a pairwise disjoint collection of measurable subsets of $X$. Use the Monotone Convergence Theorem and linearity of the integral to prove that, if $f$ is a nonnegative measurable real-valued function on $X$, $$\int_X f\ d\mu = \sum_n \int_{X_n} f\ d\mu.$$

  4. Using the Fubini/Tonelli Theorems to justify all steps, evaluate the integral $$\int_0^1 \int_y^1 x^{-3/2}\cos\left(\frac{\pi y}{2x}\right) \ dx\ dy.$$

  5. Let $I$ be the interval $[0,1]$. Let $C(I)$, $C(I\times I)$ denote the spaces of real valued continuous functions on $I$ and $I\times I$, respectively, with the usual supremum norm on these spaces. Show that the collection of finite sums of the form $$f(x,y) = \sum_i \phi_i(x)\psi_i(y) \quad \text{ (where $\phi_i, \psi_i \in C(I)$ for each $i$)}$$ is dense in $C(I\times I)$.

  6. Let $m$ be Lebesgue measure on the real line $\mathbb R$, and for each Lebesgue measurable subset $E$ of $\mathbb R$ define $$\mu(E) = \int_E \frac{1}{1+x^2}\ dm(x).$$ Show that $m$ is absolutely continuous with respect to $\mu$, and compute the Radon-Nikodym derivative $dm/d\mu$.


  1. Let $\phi(x,y) = x^2 y$ be defined on the square $S = [0,1]\times [0,1]$ in the plane, and let $m$ be two-dimensional Lebesgue measure on $S$. Given a Borel subset $E$ of the real line $\mathbb R$, define $\mu(E) = m(\phi^{-1}(E))$.
    a. Show that $\mu$ is a Borel measure on $\mathbb R$.
    b. Let $\chi_E$ denote the characteristic function of the set $E$. Show that $$\int_{\mathbb R} \chi_E , d\mu = \int_S \chi_E \circ \phi\ dm.$$ c. Evaluate the integral $\int_{-\infty}^{\infty} t^2\ d\mu(t)$.

  2. Let $f$ be a real valued and increasing function on the real line $\mathbb R$, such that $f(-\infty)=0$ and $f(\infty)=1$. Prove that $f$ is absolutely continuous on every closed finite interval if and only if $$\int_{\mathbb R} f'\ dm = 1.$$

  3. Let $F$ be a continuous linear functional on the space $L^1[-1,1]$, with the property that $F(f) = 0$ for all odd functions $f$ in $L^1[-1,1]$. Show that there exists an even function $\phi$ such that $$F(f) = \int_{-1}^1 f(x) \phi(x)\ dx,$$ for all $f\in L^1[-1,1]$. (Hint: One possible approach is to use the fact that any function in $L^p[-1,1]$ is the sum of an odd function and an even function.)