Probability with Measure
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Chapter 1 Measure Spaces and Measure
1.1 What is Measure?
Measure theory is the abstract mathematical theory that underlies all models of measurement in the real world. This includes measurement of length, area and volume, mass but also chance/probability. Measure
theory is on the one hand a branch of pure mathematics, but it also plays a key role in many applied areas such as physics and economics. In particular it provides a foundation for both the modern theory of
integration and also the theory of probability. It is one of the milestones of modern analysis and is an invaluable tool for functional analysis.
To motivate the key definitions, suppose that we want to measure the lengths of several line segments. We represent these as closed intervals of the real number line \(\R \) so a typical line segment is \([a,b]\)
where \(b > a\). We all agree that its length is \(b-a\). We write this as
\[ m([a,b]) = b-a\]
and interpret this as telling us that the measure \(m\) of length of the line segment \([a,b]\) is the number \(b-a\). We might also agree that if \([a_{1}, b_{1}]\) and \([a_{2}, b_{2}]\) are two
non-overlapping line segments and we want to measure their combined length then we want to apply \(m\) to the set-theoretic union \([a_{1}, b_{1}] \cup [a_{2}, b_{2}]\) and
\(\seteqnumber{0}{1.}{0}\)
\begin{eqnarray}
\label {firstunion} m([a_{1}, b_{1}] \cup [a_{2}, b_{2}]) & = (b_{2} - a_{2}) + (b_{1} - a_{1}) = m([a_{1}, b_{1}]) + m([a_{2}, b_{2}]).\nonumber \\ ~~&~~&~~
\end{eqnarray}
An isolated point \(c\) has zero length and so
\[ m(\{c\}) = 0.\]
and if we consider the whole real line in its entirety then it has infinite length, i.e.
\[ m(\R ) = \infty .\]
We have learned so far that if we try to abstract the notion of a measure of length, then we should regard it as a mapping \(m\) defined on subsets of the real line and taking values in the extended non-negative real
numbers \([0, \infty ]\).
Question Does it make sense to consider \(m\) on all subsets of \(\R \)?
-
Start with the interval \([0,1]\) and remove the middle third to create the set \(C_{1} = [0, 1/3)
\cup (2/3, 1]\). Now remove the middle third of each remaining piece to get \(C_{2} = [0, 1/9) \cup (2/9, 1/3) \cup (2/3, 7/9) \cup (8/9, 1]\). Iterate this process so for \(n > 2, C_{n}\) is
obtained from \(C_{n-1}\) by removing the middle third of each set within that union. The Cantor set is \(C = \bigcap _{n=1}^{\infty }C_{n}\). It turns out that \(C\) is uncountable. Does \(m(C)\) make
sense?
We’ll see later that \(m(C)\) does make sense and is a finite number (can you guess what it is?). But it turns out that there are even wilder sets in \(\R \) than \(C\) which have no length. These are quite difficult
to construct (they require the axiom of choice) so we won’t try to describe them here.
Conclusion. The set of all subsets of \(\R \) is its power set \({\cal P}(\R )\). We’ve just learned that the power set is too large to support a good theory of measure of length. So we need to find a smaller class
of subsets that we can work with.