Exercises

1. Does there exist an infinite $\sigma$ -algebra which has only countably many members?

2. Prove an analogue of Theorem 1.8 for $n$ functions.

3. Prove that if $f$ is a real function on a measurable space $X$ such that $\{x: f(x) \ge r\}$ is measurable for every rational $r$ , then $f$ is measurable.

4. Let $\{a_n\}$ and $\{b_n\}$ be sequences in $[-\infty, \infty]$ , and prove the following assertions:

provided none of the sums is of the form $\infty - \infty$ .

(c) If $a_n \le b_n$ for all $n$ , then

Show by an example that strict inequality can hold in (b).

5. (a) Suppose $f: X \to [-\infty, \infty]$ and $g: X \to [-\infty, \infty]$ are measurable. Prove that the sets

are measurable.

(b) Prove that the set of points at which a sequence of measurable real-valued functions converges (to a finite limit) is measurable.

6. Let $X$ be an uncountable set, let $\mathcal{M}$ be the collection of all sets $E \subset X$ such that either $E$ or $E^c$ is at most countable, and define $\mu(E) = 0$ in the first case, $\mu(E) = 1$ in the second. Prove that $\mathcal{M}$ is a $\sigma$ -algebra in $X$ and that $\mu$ is a measure on $\mathcal{M}$ . Describe the corresponding measurable functions and their integrals.

7. Suppose $f_n: X \to [0, \infty]$ is measurable for $n = 1, 2, 3, \dots$ , $f_1 \ge f_2 \ge f_3 \ge \dots \ge 0$ , $f_n(x) \to f(x)$ as $n \to \infty$ , for every $x \in X$ , and $f_1 \in L^1(\mu)$ . Prove that then

and show that this conclusion does not follow if the condition " $f_1 \in L^1(\mu)$ " is omitted.

8. Put $f_n = \chi_E$ if $n$ is odd, $f_n = 1 - \chi_E$ if $n$ is even. What is the relevance of this example to Fatou’s lemma?

9. Suppose $\mu$ is a positive measure on $X$ , $f: X \to [0, \infty]$ is measurable, $\int_X f d\mu = c$ , where $0 < c < \infty$ , and $\alpha$ is a constant. Prove that

Hint: If $\alpha \ge 1$ , the integrands are dominated by $\alpha f$ . If $\alpha < 1$ , Fatou’s lemma can be applied.

10. Suppose $\mu(X) < \infty$ , $\{f_n\}$ is a sequence of bounded complex measurable functions on $X$ , and $f_n \to f$ uniformly on $X$ . Prove that

and show that the hypothesis " $\mu(X) < \infty$ " cannot be omitted.

11. Show that

in Theorem 1.41, and hence prove the theorem without any reference to integration.

12. Suppose $f \in L^1(\mu)$ . Prove that to each $\epsilon > 0$ there exists a $\delta > 0$ such that $\int_E |f| d\mu < \epsilon$ whenever $\mu(E) < \delta$ .

13. Show that proposition 1.24(c) is also true when $c = \infty$ .