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Sets, Functions and Measurability - Exercises

Exercise 1

  1. Let \(\Omega\) be a state space, and let \((\mathcal{F}_i)\) be an arbitrary family of \(\sigma\)-algebra on \(\Omega\). Show that

    \[\mathcal{F}=\bigcap \mathcal{F}_i=\left\{A \subseteq \Omega\colon A \in \mathcal{F}_i \text{ for all }i\right\},\]

    is a \(\sigma\)-algebra. Conclude that for a collection \(\mathcal{C}\) of subsets of \(\Omega\),

    \[\sigma\left(\mathcal{C}\right):=\bigcap\left\{\mathcal{F}\colon \mathcal{F}\text{ is a }\sigma\text{-algebra with } \mathcal{C}\subseteq \mathcal{F}\right\},\]

    is the unique smallest \(\sigma\)-algebra containing \(\mathcal{C}\).

  2. Give an example where the union of two \(\sigma\)-algebras is not a \(\sigma\)-algebra.

  3. Let \(\Omega=\mathbb{R}\), and \(\mathcal{F}=\mathcal{B}(\mathbb{R})\) the Borel \(\sigma\)-algebra of \(\mathbb{R}\), that is, the \(\sigma\)-algebra generated by the collection \(\{O\colon O \text{ open set in }\mathbb{R}\}\). Show that \(\mathcal{B}(\mathbb{R})=\sigma(\mathcal{C}_i)\) for each \(i=1, 2\) where

    $$ \begin{aligned} \mathcal{C}1&=\left{F\colon F\text{ closed subset of }\mathbb{R}\right} & \mathcal{C}\right}}&=\left{]-\infty,b]\colon b\in \mathbb{Q

    \end{aligned} $$

  4. Let \((\Omega,\mathcal{F})\) and \((S,\mathcal{S})\) be two measurable spaces. Given a function \(X:\Omega \to S\), show that the collection of sets

    \[\sigma(X):=\left\{ X^{-1}(B)=\{\omega \in \Omega\colon X(\omega)\in B\}\colon B\in \mathcal{S} \right\},\]

    is a \(\sigma\)-algebra on \(\Omega\). Give a simple example where

    \[\left\{ X(A)=\{X(\omega)\colon \omega \in A\}\colon A\in \mathcal{F}\right\}\]

    is not a \(\sigma\)-algebra on \(S\).

Exercise 2

Let \(\Omega\) be state space and \(\mathcal{C}\) a collection of subsets of \(\Omega\), that is \(\mathcal{C} \in 2^{\Omega}\). We call \(\mathcal{C}\) a \(\pi\)-system if is stable under intersection, that is, \(A, B\) in \(\mathcal{C}\) implies \(A\cap B\) is in \(\mathcal{C}\). We call \(\mathcal{C}\) a \(\lambda\)-system if \(\emptyset \in \mathcal{C}\), \(A \in \mathcal{C}\) implies \(A^c \in \mathcal{C}\), and \(\cup A_n \in \mathcal{C}\) for any countable disjoint family \((A_n)\) of elements in \(\mathcal{C}\). Study the proof of the \(\pi\)-\(\lambda\)-system theorem of Dynkin[^1] to show the following assertions:

  1. Monotone class theorem: Suppose that \(\mathcal{C}\) is a \(\pi\)-system and \(\mathcal{H}\) a set of functions \(X:\Omega \to \mathbb{R}\) such that
    • \(1_A \in \mathcal{H}\) for every \(A\) in \(\mathcal{C}\);
    • \(X+Y \in \mathcal{H}\) and \(cX \in \mathcal{H}\) for every \(X,Y\) in \(\mathcal{H}\) and real number \(c\);
    • if \((X_n)\) is an increasing sequence of positive bounded functions converging to a bounded function \(X\), then \(X \in \mathcal{H}\). then it follows that \(\mathcal{H}\) contains all the bounded measurable functions with respect to \(\sigma(\mathcal{C})\).

  2. Let \(X:\Omega \to \mathbb{R}\) be a function. Show that any \(\sigma(X)\)-measurable random variable \(Y:\Omega \to \mathbb{R}\) can be written as

    \[Y = f(X)\]

    for a measurable function \(f:\mathbb{R}\to \mathbb{R}\) for the Borel \(\sigma\)-algebra.

Exercise 3

Let \((A_n)\) be a countable family of sets. We define the limit sup and the limit inf of the family as

\[\liminf A_n =\bigcup_{n=1}^\infty\bigcap_{k=n}^\infty A_k \quad\text{and}\quad \limsup A_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k\]

show that 1. \(\liminf A_n\subseteq \limsup A_n\) and \(\liminf A_n=\limsup A_n\) if either \(A_n\subseteq A_{n+1}\) for all \(n\) or \(A_n \supseteq A_{n+1}\) for all \(n\). 2. It holds

$$
\begin{split}
\liminf A_n &=\left\{ x \in X\colon x\text{ is in all but finitely many } A_n \right\}\\
\limsup A_n &=\left\{ x \in X\colon x\text{ is in infinitely many } A_n \right\}\\
\end{split}
$$
  1. \(P[\liminf A_n]\leq \liminf P[A_n]\leq \limsup P[A_n]\leq P[\limsup A_n]\) and give an example for which all inequalities are strict.
  2. if \(\sum P[A_n]<\infty\), then \(P[\limsup A_n]=0\).

Exercise 4

probability space. Let \(X:\Omega \to \mathbb{R}\) be a random variable. Define

\[F(t):=P[X\leq t], \quad t \in \mathbb{R}.\]

which is called the cumulative distribution function of \(X\). Show that 1. \(F:\mathbb{R}\to \mathbb{R}\) is increasing, \(\lim_{t \to -\infty}F(t)=0\), \(\lim_{t \to \infty}F(t)=1\) and \(F\) is right-continuous.[^2] 2. \(F\) is measurable for the Borel \(\sigma\)-algebra; 3. \(F\) has at most countably many discontinuous points.

Exercise 5

  1. Find a sequence of positive random variables \((X_n)\) such that \(E[X_n]\to 0\) but \(P[\limsup X_n>\liminf X_n]=1\), that is \(X_n\) converges \(P\)-almost nowhere.
  2. Find a sequence of positive random variables \((X_n)\) such that \(X_n\to X\) \(P\)-almost surely and in \(L^1\), but \(\sup_n X_n\) is not integrable.
  3. Show that if \(X_n\to X\) in \(L^1\), then \(X_n\to X\) in probability.[^3] Find an example such that the reciprocal is not true.
  4. Show that the dominated convergence theorem holds when instead of requiring \(X_n\to X\) \(P\)-almost surely, on suppose that \(X_n\to X\) in probability.
  5. Let \(\alpha\geq 1\) and \(X\) be an integrable positive random variable. Show that \(\lim E[n\ln(1+(X/n)^\alpha)]\) exists and compute its value.[^4]