Simple Functions

1.16 Definition A complex function $s$ on a measurable space $X$ whose range consists of only finitely many points will be called a simple function. Among these are the nonnegative simple functions, whose range is a finite subset of $[0, \infty)$ . Note that we explicitly exclude $\infty$ from the values of a simple function.

If $\alpha_1, \dots, \alpha_n$ are the distinct values of a simple function $s$ , and if we set $A_i = \{x: s(x) = \alpha_i\}$ , then clearly

where $\chi_{A_i}$ is the characteristic function of $A_i$ , as defined in Sec. 1.9(d).

It is also clear that $s$ is measurable if and only if each of the sets $A_i$ is measurable.

1.17 Theorem Let $f: X \to [0, \infty]$ be measurable. There exist simple measurable functions $s_n$ on $X$ such that

(a) $0 \le s_1 \le s_2 \le \dots \le f$ .

(b) $s_n(x) \to f(x)$ as $n \to \infty$ , for every $x \in X$ .

PROOF Put $\delta_n = 2^{-n}$ . To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t < (k+1)\delta_n$ . Define

Each $\varphi_n$ is then a Borel function on $[0, \infty]$ ,

$0 \le \varphi_1 \le \varphi_2 \le \cdots \le t$ , and $\varphi_n(t) \to t$ as $n \to \infty$ , for every $t \in [0, \infty]$ . It follows that the functions

satisfy (a) and (b); they are measurable, by Theorem 1.12(d).

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