Ex4-2

Solution to Chapter 4, Exercise 2

''Question: Let $$f$$be a simple function, taking its distinct values on disjoint sets $$E_1, \cdots, E_N$$. Show that $$f$$is measurable if and only if $$E_1,\cdots,E_N$$are measurable.''

Solution: Let the distinct values be $$a_1,\cdots,a_N$$so that

$$f(x)=\sum_{k=1}^N a_k\chi_{E_k}(x)$$

on $$E=\bigcup_{k=1}^N E_k$$.

Suppose $$f$$is measurable. Then for any $$a$$, $$\{f \geq a\}$$and $$\{f>a\}$$are measurable. Then

$$\{f=a\}=\{f \geq a\}-\{f>a\}$$

is measurable for any $$a$$. Since $$E_k=\{f=a_k\}$$, each $$E_k$$is measurable.

Conversely, suppose $$E_1, \cdots, E_N$$are measurable. Then for any $$a$$, if $$a \geq \max\{a_1,\cdots,a_N\}$$, then

$$\{f>a\}=\emptyset$$

is measurable so $$f$$is measurable. If $$a<\max\{a_1,\cdots,a_N\}$$, then

$$\{f>a\}=\bigcup_{k~:~a_k>a}E_k$$

is a finite union of measurable sets so it is measurable. Thus, $$f$$is measurable.