Probability with Measure

Chapter 2 Real Analysis

In this chapter we cover a small amount of content that would (mostly) fit more naturally within a course on real analysis, but which was not covered as part of earlier courses due to lack of time. We’ll first put together a natural way of dealing with limits that might take the values \(\pm \infty \), and then we’ll think about convergence of sequences of functions.

Given a set \(A\sw \R \), or more generally a subset \(A\sw S\) for some measurable space \((S,\Sigma )\), we define the indicator function \(\1_A:S\to \R \) by

\begin{equation*} \1_A(x)= \begin{cases} 1 & \text { if }x\in A \\ 0 & \text { if }x\notin A. \\ \end {cases} \end{equation*}

We will study indicator functions a little in this chapter, in Exercise 2.5. They will become very important to us in Chapters 3 and 4.

2.1 The extended reals

The extended reals are the real numbers, with \(\pm \infty \) included,

\[\ov {\R }=\R \cup \{-\infty ,\infty \}.\]

We have already seen that measures take values in \([0,\infty ]\), and we’ll need to learn how to deal naturally \(\pm \infty \) within this course. Let’s recap a few key definitions.

You should remember the definition of a convergent sequence \(a_n\to a\) of real numbers:

\begin{equation*} \forall \eps >0\;\stackrel {\text {`eventually'}}{\overbrace {\exists N\in \N \;\forall n\geq N,}}\; |a_n-a|\leq \eps . \end{equation*}

The best way to understand this is definition is by thinking about the terms marked as the word ‘eventually’. By eventually we mean that, if you look far enough down the sequence, we will see some event happen for all remaining terms. So convergence means that for all \(\eps >0\), eventually \(|a_n-a|\leq \eps \). Note that we can use \(<\eps \) or \(\leq \eps \) here – it is straightforward to show that these give equivalent definitions.

We also need to think about limits that have the value \(\pm \infty \). In real analysis we might call this ‘divergence’ because \(\pm \infty \) are not elements of \(\R \), but when we work within the extended real numbers \(\ov {\R }=\R \cup \{\infty ,-\infty \}\) we can also use the term ‘convergence to \(\pm \infty \)’ for this case. We won’t be fussy about that point of language within this course. The definition of \(a_n\to \infty \) is that

\[\forall M\in (0,\infty )\;\exists N\in \N \;\forall n\geq N,\; a_n\geq M.\]

In words, for all \(M\in (0,\infty )\), eventually \(a_n\geq M\). For the case \(a_n\to -\infty \) we use \(M\in (-\infty ,0)\) and require instead that \(a_n\leq M\).

You already know that a bounded monotone sequence of real numbers converges. Working in \(\ov {\R }\) allows us to remove the boundedness requirement.

Proof: Without loss of generality we can assume that \((a_n)\) is monotone increasing i.e. \(a_{n+1}\geq a_n\) for all \(n\in \N \), or else we could consider \((-a_n)\) in place of \((a_n)\). Note that in real analysis you have already proved the case where \((a_n)\) is a bounded sequence. If \((a_n)\) is unbounded and increasing then \(\sup _n a_n=\infty \), so for any \(M\in (0,-\infty )\) there exists \(N\) with \(a_N\geq M\), which implies that \(M\leq a_N\leq a_{N+1}\leq a_{N+2}\leq \ldots \). In words, eventually \(a_n\geq M\). Hence \(a_n\to \infty \).   ∎

We can do arithmetic in \(\ov {\R }\) in natural ways, for example if \(a\in \R \) then \(a+\infty =\infty \). We have to be careful though, some objects like \(\infty -\infty \) and \(\frac {\infty }{\infty }\) do not make sense (formally, they are undefined) but otherwise it works as you’d expect. Such restrictions are necessary. They exist to prevent nonsensical calculations like \(1=\frac {\infty }{\infty }=\frac {\infty +\infty }{\infty }=\frac {\infty }{\infty }+\frac {\infty }{\infty }=1+1=2\). You can find the precise rules for arithmetic in \(\ov {\R }\) in Section 0.2.

2.1.1 The Borel \(\sigma \)-field and Lebesgue measure on \(\ov {\R }\)

Recall that we defined the Borel \(\sigma \)-field \(\mc {B}(\R )\) in Section 1.4. We extend the Borel \(\sigma \)-field to \(\ov {\R }\) by defining

\[\mc {B}(\ov {\R })=\sigma (\mc {B}(\R ),\{\infty \},\{-\infty \}),\]

that is the smallest \(\sigma \)-field that contains all elements of \(\mc {B}(\R )\) and the singleton sets \(\{\infty \}\) and \(\{-\infty \}\). The following lemma summarizes the connection.

Proof: We’ll prove the forwards and backwards implications in turn. We’ll start with the reverse implication. Let \(A\sw \ov {\R }\) and assume that \(A\cap \R \in \mc {B}(\R )\). We can write

\[A=(A\cap \R )\cup (A\cap \{\infty \})\cup (A\cap \{-\infty \}).\]

Note that \(A\cap \{\infty \}\) is either empty or equal to \(\{\infty \}\), which in either case is an element of \(\mc {B}(\ov {\R })\). Similarly, \(A\cap \{-\infty \}\in \mc {B}(\ov {\R })\). Hence \(A\in \mc {B}(\ov {\R })\), as required.

For the forwards implication, define

\begin{equation} \label {eq:ABR_breakdown} \Sigma =\mc {B}(\R ) \cup \Big \{A\cup \{\infty \}\-A\in \mc {B}(\R )\Big \} \cup \Big \{A\cup \{-\infty \}\-A\in \mc {B}(\R )\Big \} \cup \Big \{A\cup \{-\infty ,\infty \}\-A\in \mc {B}(\R )\Big \}. \end{equation}

It is straightforward (but tedious) to check that \(\Sigma \) is a \(\sigma \)-field on \(\ov {\R }\). We have \(\{\infty \}\in \Sigma \) and \(\{-\infty \}\in \Sigma \), and for all \(A\in \mc {B}(\R )\) we have \(A\in \mc {B}(\ov {\R })\). Hence, using that \(\mc {B}(\ov {\R })\) is the smallest \(\sigma \)-field generated by such sets, we have \(\mc {B}(\ov {\R })\sw \Sigma \). In particular, if \(A\in \mc {B}(\ov {\R })\) then \(A\in \Sigma \), which means that \(A\) is within one of the sets making up the right hand side of (2.1). In all four such cases we have \(A\cap \R \in \mc {B}(\R )\), as required.   ∎

Recall that we defined Lebesgue measure \(\lambda (A)\) for \(A\in \mc {B}(\R )\) in Section 1.5. We extend Lebesgue measure to \(A\in \mc {B}(\ov {\R })\) by setting

\begin{equation} \label {eq:ovR_Leb} \lambda (A)=\lambda (A\cap \R ). \end{equation}

Lemma 2.1.3 ensures that \(A\cap \R \in \mc {B}(\R )\), so the right hand side of (2.2) may be used to define the left. In words, we don’t put any weight at \(\pm \infty \). It is straightforward to check that this gives a measure on \((\ov {\R },\mc {B}(\R ))\), in very similar style to part (b) of Exercise 1.5.