1991 Nov 21

1. a. Let $f_n$ be a sequence of continuous, real valued functions on $[0, 1]$ which converges uniformly to $f$. Prove that $\lim_{n\to \infty} f_n (x_n) = f(1/2)$ for every sequence $\{x_n\}$ that converges to $1/2$.
b. Must the conclusion still hold if the convergence is only point-wise? Explain.

2. Let $f \colon \mathbb R \to \mathbb R$ be differentiable and assume there is no $x \in \mathbb R$ such that $f(x) = f'(x) = 0$. Show that $S = \{x : 0 \leq x \leq 1, f(x) = 0\}$ is finite.

3. If $(X, \Sigma, \mu)$ is a measure space and if $f$ is $\mu$-integrable, show that for every $\epsilon > 0$ there is $E\in \Sigma$ such that $\mu(E) < \infty$ and $$\int_{X\setminus E} |f| \ d\mu < \epsilon.$$

4. If $(X, \Sigma, \mu)$ is a measure space, $f$ is a non-negative measurable function, and $\nu(E) = \int_E f\ d\mu$, show that $\nu$ is a measure.

5. Suppose $f$ is a bounded, real valued function on $[0, 1]$. Show that $f$ is Lebesgue measurable if and only if $$\sup \int \psi\ dm = \inf \int \phi\ dm,$$ where $m$ is Lebesgue measure and $\phi$ and $\psi$ range over all simple functions for which $\psi \leq f \leq \phi$.

6. If $f$ is Lebesgue integrable on $[0, 1]$ and $\epsilon > 0$, show that there is $\delta > 0$ such that for all measurable sets $E \subset [0, 1]$ with $m(E) < \delta$, $$\left|\int_E f\ dm\right| < \epsilon$$ (cf. 1992 Apr Prob 4, 1997 Nov Prob 6, 2003 Apr Prob 4).

7. Suppose $f$ is a bounded, real valued, measurable function on $[0, 1]$ such that $\int x^n f\ dm = 0$ for $n = 0, 1, 2, \dots$, with $m$ Lebesgue measure. Show that $f(x) = 0$ a.e. (cf. 1992 Apr Prob 6, 1992 Nov Prob 7, 1995 Nov Prob 6, 1996 Nov Prob B).

8. If $\mu$ and $\nu$ are finite measures on the measurable space $(X, \Sigma)$, show that there is a nonnegative measurable function $f$ on $X$ such that for all $E$ in $\Sigma$, $$\int_E (1 - f)\ d\mu = \int_E f\ d\nu.$$ (cf. 1997 Nov Prob 7).

9. If $f$ and $g$ are integrable functions on $(X, S, μ)$ and $(Y, T , ν)$, respectively, and $F(x, y) = f (x) g(y)$, show that $F$ is integrable on $X \times Y$ and $$\int F d(\mu \times \nu) = \int f \ d\mu \int g\ d\nu.$$