Probability with Measure

0.2 Preliminaries

This section contains lots of definitions, mostly from earlier courses, that we will use later on. It should be familiar to you but there may be one or two minor extensions of ideas you have seen before.

• 1. Set Theory.

  Let \(S\) be a set, with subsets \(A,B\) and \(A_n\).

  Complement: \(A^{c} = \{x \in S; x \notin A\}.\)

  Union: \(A \cup B = \{x \in S; x \in A~\mbox {or}~x \in B\}\).

  Intersection: \(A \cap B = \{x \in S; x \in A~\mbox {and}~x \in B\}\).

  Set theoretic difference: \(A \sc B = A \cap B^{c}\).

  Finite unions and intersections: \(\bigcup _{i=1}^{n} A_{i} = A_{1} \cup A_{2} \cup \ldots \cup A_{n}\) and \(\bigcap _{i=1}^{n} A_{i} = A_{1} \cap A_{2} \cap \ldots \cap A_{n}.\)

  More generally, if \(I\) is some set and \(A_i\sw S\) for all \(i\in I\) then we define

  \[ \bigcup _{i\in I} A_i=\{x \- x\in A_i\text { for some }i\in I\} \qquad \bigcap _{i\in I} A_i=\{x \- x\in A_i\text { for all }i\in I\}. \]

  Countable unions and intersections are precisely the case \(I=\N \), usually written as \(\bigcup _{i=1}^{\infty } A_{i}\) and \(\bigcup _{i=1}^{\infty } A_{i}\).

  De Morgan’s laws state that:

  \[ S\sc \left (\bigcap _{i\in I} A_{i}\right ) = \bigcup _{i\in I} S\sc A_{i},\]

  \[ S\sc \left (\bigcup _{i\in I} A_{i}\right ) = \bigcap _{i\in I} S\sc A_{i}.\]

  The Cartesian product of sets \(S\) and \(T\) is the set \(S \times T = \{(s,t)\- s \in s, t \in T\}.\)

• 2. Sets of Numbers

  • • Natural numbers \(\N = \{1, 2,3, \ldots \}\).

  • • Non-negative integers \(\Z _{+} = \N \cup \{0\} = \{0,1, 2,3, \ldots \}\).

  • • Integers \(\Z \).

  • • Rational numbers \(\mathbb {Q}\).

  • • Real numbers \(\R \).

  • • Complex numbers \(\C \).

  A set \(X\) is countable if there exists an injection between \(X\) and \(\N \). A set is uncountable if it fails to be countable. \(\N , \Z _{+}, \Z \) and \(\Q \) are countable. \(\R \) and \(\C \) are uncountable. All finite sets are countable.

• 3. Images and Preimages.

  Suppose that \(S_{1}\) and \(S_{2}\) are two sets and that \(f:S_{1} \rightarrow S_{2}\) is a mapping (or function). Suppose that \(A \subseteq S_{1}\). The image of \(A\) under \(f\) is the set \(f(A) \subseteq S_{2}\) defined by

  \[ f(A) = \{y \in S_{2}; y = f(x)~\mbox {for some}~x \in S_{1}\}.\]

  If \(B \subseteq S_{2}\) the inverse image or pre-image of \(B\) under \(f\) is the set \(f^{-1}(B) \subseteq S_{1}\) defined by

  \[ f^{-1}(B) = \{x \in S_{1} ; f(x) \in B\}.\]

  Note that \(f^{-1}(B)\) makes sense irrespective of whether the mapping \(f\) is invertible.

  Key properties are, with \(A, A_{1}, A_{2} \subseteq S_{1}\) and \(B, B_{1}, B_{2} \subseteq S_{2}\) :

  \[\begin {aligned} f^{-1}(B_{1} \cup B_{2}) &= f^{-1}(B_{1}) \cup f^{-1}(B_{2}),\\ f^{-1}(B_{1} \cap B_{2}) &= f^{-1}(B_{1}) \cap f^{-1}(B_{2}),\\ f^{-1}(A^{c}) &= f^{-1}(A)^{c},\\ f(A_{1} \cup A_{2}) &= f(A_{1}) \cup f(A_{2}),\\ f(A_{1} \cap A_{2}) &\subseteq f(A_{1}) \cap f(A_{2}),\\ \end {aligned}\]

  Note also that if \(A\sw B\) then \(f(A)\sw f(B)\) and \(f^{-1}(A)\sw f^{-1}(B)\).

• 4. Extended Real Numbers

  We will often find it convenient to work with \(\infty \) and \(-\infty \). These are not real numbers, but we find it convenient to treat them a bit like real numbers. To do so we specify some extra arithmetic rules:

  • • for all \(x\in \R \) we have \(\infty + x = x + \infty = \infty ,\)

  • • for \(x>0\) we have \(x\times \infty = \infty \times x = \infty \),

  • • for all \(x\in \R \) we have \(\frac {x}{\infty } = 0\) and \(\frac {\infty }{x}=\infty ,\)

  • • \(\infty \times (-1) = -\infty \) and \((-\infty )\times (-1)=\infty .\)

  Combining these rules and using the usual properties of real arithmetic (e.g. \(a\times b=b\times a\)) allows us to deduce further properties, for example for \(x<0\) we have \(x\times \infty = (-1)\times (-x) \times \infty = (-1)\times \infty = -\infty \). Any arithmetic expressions involving \(\pm \infty \) that are not specified by the above rules are undefined. In particular, \(\infty - \infty \), \(0\times \infty \) and \(\frac {\infty }{\infty }\) are undefined.

  We write \(\ov {\R } = \{-\infty \} \cup \R \cup \{\infty \}\), which is known as the extended real numbers. We also specify that, for all \(x\in \R \),

  \[-\infty < x < \infty .\]

• 5. Analysis.

  • • sup and inf. If \(A\) is a bounded set of real numbers, we write \(\sup (A)\) and \(\inf (A)\) for the real numbers that are their least upper bounds and greatest lower bounds (respectively.) If \(A\) fails to be bounded above, we write \(\sup (A) = \infty \) and if \(A\) fails to be bounded below we write \(\inf (A) = - \infty \). Note that \(\inf (A) = -\sup (-A)\) where \(-A = \{-x \- x \in A\}\). If \(f: S \rightarrow \R \) is a mapping, we write \(\sup _{x \in S}f(x)= \sup \{f(x); x \in S\}\). A very useful inequality is

    \[ \sup _{x \in S}|f(x) + g(x)| \leq \sup _{x \in S}|f(x)| + \sup _{x \in S}|g(x)|.\]

  • • Sequences and Limits. Let \((a_{n}) = (a_{1}, a_{2}, a_{3}, \ldots )\) be a sequence of real numbers. It converges to the real number \(a\) if given any \(\eps > 0\) there exists a natural number \(N\) so that whenever \(n > N\) we have \(|a - a_{n}| < \eps \). We then write \(a = \lim _{n \rightarrow \infty }a_{n}\).

    A sequence \((a_{n})\) which is monotonic increasing (i.e. \(a_{n} \leq a_{n+1}\) for all \(\nN \)) and bounded above (i.e. there exists \(K > 0\) so that \(a_{n} \leq K\) for all \(\nN \)) converges to \(\sup _{\nN }a_{n}\).

    A sequence \((a_{n})\) which is monotonic decreasing (i.e. \(a_{n+1} \leq a_{n}\) for all \(\nN \)) and bounded below (i.e. there exists \(L > 0\) so that \(a_{n} \geq L\) for all \(\nN \)) converges to \(\inf _{\nN }a_{n}\).

    A subsequence of a sequence \((a_{n})\) is itself a sequence of the form \((a_{r_n})\) where \(r_{n}<r_{n+1}\) for all \(n\in \N \).

  • • Series. If the sequence \((s_{n})\) converges to a limit \(s\) where \(s_{n} = a_{1} + a_{2} + \cdots + a_{n}\) we write \(s = \sum _{n=1}^{\infty }a_{n}\) and call it the sum of the series. If each \(a_{n} \geq 0\) then the sequence \((s_{n})\) is either convergent to a limit or properly divergent to infinity. In the latter case we write \(s = \infty \) and interpret this in the sense of extended real numbers.

  • • Continuity. A function \(f:\R \rightarrow \R \) is continuous at \(a \in \R \) if given any \(\eps > 0\) there exists \(\de > 0\) so that \(|x - a| < \de \Rightarrow |f(x) -f(a)| < \eps \). Equivalently \(f\) is continuous at \(a\) if given any sequence \((a_{n})\) that converges to \(a\), the sequence \((f(a_{n}))\) converges to \(f(a)\).

    \(f\) is a continuous function if it is continuous at every \(a \in \R \).