Probability with Measure

3.2 Borel measurable functions

To make use of the various parts of Theorem 3.1.5 we need some functions that we already know are measurable. Example 3.1.2 is a good start and you’ll find some more examples within the exercises at the end of this chapter. In the special case of functions \(f:\R \to \R \) the following lemma is very useful.

Lemma 3.2.1 If \(f:\R \to \R \) is continuous then \(f\) is Borel measurable.

Proof: The proof is in Section 3.3, which is marked with a \((\Delta )\) for independent study. ∎

The most important fact to realize about functions \(f:\R \to \R \) is that, although it is possible to construct non-measurable examples, essentially all examples of functions \(f:\R \to \R \) that we encounter in practice are Borel measurable. We have now set up all these tools to prove this, for the functions that we commonly use. There are several examples in Exercise 3.1. Here’s one more.

Example 3.2.2 The function

\[ f(x)=\begin {cases} \sin x & \text { if } x\in [\frac {-\pi }{2},\frac {\pi }{2}], \\ 0 & \text { otherwise}, \end {cases} \]

is Borel measurable. To see this, note that the function \(\sin \) is continuous on \(\R \), and hence measurable by Lemma 3.2.1, which is a good start. To make the link to \(f\) we note that

\[f(x)=\1_{[\frac {-\pi }{2},\frac {\pi }{2}]}(x)\sin (x).\]

Example 3.1.2 gives that \(\1_{[-\frac {\pi }{2},\frac {\pi }{2}]}\) is measurable, because intervals are Borel sets. Theorem 3.1.5 (in particular, the multiplication part) then gives that \(f\) is Borel measurable.

We could reach the same conclusion in many different ways. For example, the function

\[ g(x)=\begin {cases} -1 & \text { if }x<\frac {-\pi }{2}, \\ \sin x & \text { if } x\in [\frac {-\pi }{2},\frac {\pi }{2}], \\ 1 & \text { if }x>\frac {\pi }{2}, \end {cases} \]

is continuous and hence measurable by Lemma 3.2.1. We can write \(f\) as

\[f(x)=g(x)-\1_{(\frac {\pi }{2},\infty )}(x)+\1_{(-\infty ,\frac {-\pi }{2})}(x)\]

and Theorem 3.1.5 (this time, the addition part) tells us that \(f\) is Borel measurable.