Probability with Measure
Chapter 3 Measurable Functions
In this chapter we restrict ourselves to studying a particular kind of function, known as a measurable function. For measure theory, this is an important step, because it allows us to exclude some very strangely behaved examples that would disrupt our theory.
More specifically, in Section 1.6 we saw the existence subsets of \(\R \) that were not measurable, with respect to Lebesgue measure. These sets had no meaningful concept of ‘length’. It is clear that integration is closely connected to length, for example integrating the indicator function
\(\seteqnumber{0}{3.}{0}\)\begin{equation*} \1_A(x)= \begin{cases} 1 & \text { if }x\in A \\ 0 & \text { if }x\notin A \\ \end {cases} \end{equation*}
in the case \(A=[a,b]\) using a naive ‘area under the curve’ approach gives \(\int _\R \1_A(x)\,dx=b-a=\lambda (A)\). If \(A\) is non-measurable then integrating the function \(\1_A\) would be dangerously close to trying to measure the length of \(A\), which we know we cannot do. We conclude that, just as with sets, we need a similar concept of a measurable function.
3.1 Overview
In this section, we deal with functions \(f:S\to \R \), where \((S,\Sigma )\) is a measurable space. The key definition is the following.
We will sometimes wish to restrict to the case \(S=\R \), in which case we would normally take \(\Sigma =\mc {B}(\R )\) as the Borel sets. In this case, measurable functions \(f:\R \to \R \) are often known as Borel measurable.
We can’t yet explain exactly why Definition 3.1.1 is a sensible class of functions to be interested in. This will become clear in Section 4.1 when we begin to construct the Lebesgue integral. Note that the \(\sigma \)-field \(\Sigma \) is the key object in Definition 3.1.1. If we were to use a different \(\sigma \)-field \(\Sigma \) on the same set \(S\), then we might change whether \(f:S\to \R \) was measurable.
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Example 3.1.2 Let \(X\in \Sigma \). Then \(\1_X\) is measurable. To see this let us write \(f=\1_X\) and note that \(f(x)\) takes the values \(0\) if \(x\in X\) and \(1\) if \(x\notin X\). Therefore we have
\(\seteqnumber{0}{3.}{0}\)\begin{equation*} f^{-1}(A)= \begin{cases} \emptyset & \text { if } 0\notin A\text { and }1\notin A \\ S & \text { if } 0\in A\text { and }1\in A \\ X & \text { if } 0\notin A\text { and }1\in A \\ S\sc X & \text { if } 0\in A\text { and }1\notin A. \end {cases} \end{equation*}
In all cases we have that \(f^{-1}(A)\in \Sigma \).
A interesting special case is provided the indicator function \(\1_\Q \) of the rational numbers, defined from \(\R \to \R \). This function is discontinuous at all points, because between any pair of rationals there is an irrational number, and vice versa. Since \(\Q \in \mc {B}(\R )\), the function \(\1_\Q \) is Borel measurable.
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Example 3.1.3 \((\star )\) It is possible to construct non-measurable functions, for example if we take \((S,\Sigma )=(\R ,\mc {B}(\R )\) and define \(f:\R \to \R \) by \(f(x)=\1_{\mathscr {V}}(x)\) where \(\mathscr {V}\) is the non-measurable set constructed in Section 1.6. Then \(f^{-1}(\{1\})=\mathscr {V}\notin \mc {B}(\R )\).
It is usually impossible to use Definition 3.1.1 to check directly that some function \(f\) is measurable, because \(\mc {B}(\R )\) is too big to check \(f^{-1}(A)\) for all \(A\in \mc {B}(\R )\). In fact it is sufficient to check a much smaller class of subsets, for example using the following result.
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Lemma 3.1.4 Let \(f:S\to \R \). Then the following statements are equivalent.
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1. \(f\) is measurable.
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2. \(f^{-1}((a,\infty ))\in \Sigma \) for all \(a\in \R \).
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3. \(f^{-1}([a,\infty ))\in \Sigma \) for all \(a\in \R \).
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4. \(f^{-1}((-\infty ,a))\in \Sigma \) for all \(a\in \R \).
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5. \(f^{-1}((-\infty ,a])\in \Sigma \) for all \(a\in \R \).
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Proof: It is immediate that part 1 \(\ra \) all of the other parts. We will show here that parts 2, 3, 4 and 5 are all equivalent to each other. Proof that these (all) imply part 1 can be found in Section 3.3, which is marked with a \((\Delta )\) for independent study.
Note first that part 2 \(\iff \) part 5, as \(f^{-1}(A)^{c} = f^{-1}(A^{c})\) and \(\Sigma \) is closed under taking complements. The fact that part 3 \(\iff \) part 4 is proved similarly. To see that part 2 \(\ra \) part 3 we use that \([a, \infty ) = \bigcap _{n=1}^{\infty }(a-1/n, \infty )\) and so
\[ f^{-1}([a, \infty )) = \bigcap _{n=1}^{\infty }f^{-1}((a-1/n, \infty ))\]
and the result follows since \(\Sigma \) is closed under countable intersections. Similarly, to see that part 3 \(\ra \) part 2 we use that
\[ f^{-1}((a, \infty )) = \bigcup _{n=1}^{\infty }f^{-1}([a+1/n, \infty ))\]
and the fact that \(\Sigma \) is closed under countable unions. We thus have part part 5 \(\iff \) part 2 \(\iff \) part 3 \(\iff \) part 4. ∎
There is nothing special about half-open intervals in Lemma 3.1.4. Lots of other types of subset of \(\R \) will do and we will encounter a few more within this course, such as in Exercise 3.2, or Lemma 3.3.5 within the independent reading.
Lemma 3.1.4 provides a direct way of showing that functions are measurable, but an indirect way is often easier: we can use measurable functions to construct more measurable functions. In fact, nearly everything that combines measurable functions together will create more measurable functions – much like the situation we already established for measurable sets. There are also similarities to the algebra of limits from real analysis, which gives the next theorem its name.
Recall the ‘pointwise’ notation for functions that we introduced in Section 2.3 e.g. \(f+g\) means the function with values \((f+g)(x)=f(x)+g(x)\).
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Theorem 3.1.5 (Algebra of measurable functions) Let \(f,g:S\to \R \) be measurable and let \(\alpha \in \R \). The following functions are measurable:
\(\seteqnumber{0}{3.}{0}\)\begin{equation} \label {eq:pointwise_1} f+g, \qquad fg, \qquad \alpha f, \qquad 1/f, \qquad f\vee g, \qquad f\wedge g. \end{equation}
In the case of \(1/f\) we must assume \(f(x)\neq 0\) for all \(x\in S\).
If \(f_n:S\to \R \) for all \(\nN \) then the following functions are measurable:
\(\seteqnumber{0}{3.}{1}\)\begin{equation} \label {eq:pointwise_2} \inf _n f_n \qquad \sup _n f_n \qquad \liminf _{n\to \infty }f_n, \qquad \limsup _{n\to \infty } f_n, \qquad \lim _{n\to \infty } f_n \end{equation}
In the case of \(\lim _{n} f_n\) we must assume that the limit exists (pointwise).
If \(G:\R \to \R \) is Borel measurable then \(G\circ f\), defined by \((G\circ f)(x)=G(f(x))\), is measurable.
Proof: The proof is in Section 3.4, which is marked with a \((\Delta )\) for independent study. ∎
The functions in (3.1) are defined pointwise, as introduced in Section 2.3. The functions in (3.2) are also defined pointwise, for example \((\inf _n f_n)(x)=\inf _n f_n(x)\), and similarly for the others. Note that in the case of \(\lim \), the function \((\lim _n f_n)(x)= \lim _n f_n(x)\) is only defined if the limit exists for all \(x\). Strictly, at this point we treat only real valued functions so we should require that the \(\inf \)s, \(\sup \)s, and so on are not \(\pm \infty \), but we will remove this restriction in Section 3.6.