III. Measurable transformations and random variables

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Find these notes at gregorkb.github.io/probtheory/

Measurable transformations

• Measurable mapping: Let \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\) be measurable spaces. A mapping \(T:\Omega_1 \to \Omega_2\) is called \(\langle \mathcal{F}_1,\mathcal{F}_2 \rangle\)-measurable if \(T^{-1}(A) \equiv \{\omega : T(\omega) \in A\} \in \mathcal{F}_1\) for all \(A \in \mathcal{F}_2\).

  We could say “function” instead of “mapping”, because these mean the same thing.

• Examples: Let \(\Omega_1 = \Omega_2 = \Omega = \{a,b,c,d\}\) and \(\mathcal{F}_1 = \{\Omega, \emptyset,\{a\},\{b,c,d\}\}\) and \(\mathcal{F}_2 = \mathcal{P}(\Omega)\). Check measurability of mappings \[ \begin{align} T_1(\omega) &\equiv a \text{ for all }\omega \in \Omega \\ T_2(\omega) & \equiv \left\{\begin{array}{ll} a& \text{ if } \omega = a,c\\ c& \text{ if } \omega = c,d.\end{array}\right. \end{align} \]

  This is Example 2.1.1 on page 40 of A&L.

• Result to ease checking measurability: Let \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\) be measurable spaces such that \(\mathcal{F}_2 = \sigma \langle \mathcal{C}\rangle\) for some collection \(\mathcal{C}\subset \mathcal{P}(\Omega_2)\). Show that \(T:\Omega_1\to \Omega_2\) is \(\langle \mathcal{F}_1,\mathcal{F}_2\rangle\)-measurable if \(T^{-1}(A) \in \mathcal{F}_1\) for all \(A \in \mathcal{C}\).

• Composition of measurable maps: Let \((\Omega_i,\mathcal{F}_i)\), \(i=1,2,3\) be measurable spaces and let \(T_1:\Omega_1 \to \Omega_2\) and \(T_2: \Omega_2 \to \Omega_3\) be mappings which are \(\langle \mathcal{F}_1,\mathcal{F}_2 \rangle\)- and \(\langle \mathcal{F}_2,\mathcal{F}_3 \rangle\)-measurable, respectively. Show that the mapping \(T:\Omega_1 \to \Omega_3\) given by the composition \((T_2 \circ T_1)(\omega) \equiv T_2(T_1(\omega))\) for \(\omega \in \Omega_1\) is \(\langle \mathcal{F}_1,\mathcal{F}_3\rangle\)-measurable.

  See Proposition 2.1.1 on page 41 of A&L.

• Application: Let \((\Omega,\mathcal{F})\) be a measurable space and let \(T:\Omega \to \mathbb{R}\) be a mapping such that \(T^{-1}((a,b]) \in \mathcal{F}\) for all \((a,b] \subset \mathbb{R}\).
  Argue that \(T\) is \(\langle \mathcal{F},\mathcal{B}(\mathbb{R})\rangle\)-measurable.

• Special case of measurability: Given a measurable space \((\Omega,\mathcal{F})\), a mapping \(f:\Omega \to \mathbb{R}\) is simply called measurable if it is \(\langle \mathcal{F},\mathcal{B}(\mathbb{R})\rangle\)-measurable.

• Measurable function: Given a measurable space \((\Omega,\mathcal{F})\), a function \(f :\Omega \to \mathbb{R}\) is measurable if \(f^{-1}((-\infty,x]) \equiv \{\omega : f(\omega) \leq x\} \in \mathcal{F}\) for each \(x\in \mathbb{R}\).

  This definition works owing to \(\mathcal{B}(\mathbb{R}) = \sigma \langle \{(-\infty,x], x \in \mathbb{R}\} \rangle\).

• Random variable: Let \((\Omega,\mathcal{F},P)\) be a probability space. Then a function \(X: \Omega \to \mathbb{R}\) is a random variable if it is measurable, i.e. if \(X^{-1}((-\infty,a]) \equiv \{\omega : X(\omega) \leq a\} \in \mathcal{F}\) for each \(a \in \mathbb{R}\).

• Measurability of a continuous function: In the measurable space \((\mathbb{R},\mathcal{B}(\mathbb{R}))\), let \(f:\mathbb{R}\to \mathbb{R}\) be a continuous function. Show that \(f\) is measurable.

  See Proposition 2.1.2 of A&L. Recall: A function \(f:\mathbb{R}\to \mathbb{R}\) is continuous at \(x\) if for every \(\epsilon > 0\) there exists a \(\delta(\epsilon) > 0\) such that \(f(y) \in (f(x) - \epsilon,f(x) + \epsilon)\) for all \(y \in (x - \delta(\epsilon),x+\delta(\epsilon))\). A function \(f\) is continuous on a set \(D \subset \mathbb{R}\) if it is continuous at all \(x \in D\). See definitions on pages 77 and 93 of Kirkwood. The result can be extended to continuous functions \(f:\mathbb{R}^k \to \mathbb{R}^p\).

• Check whether the indicator function \[ 1_{(-\infty,c]}(x) \equiv \left\{\begin{array}{ll} 1,& x \leq c\\ 0,&x > c\end{array}\right. \] is measurable.

• Compositions of measurable functions: Let \((\Omega,\mathcal{F})\) be a measurable space and let \(f_1,\dots,f_k : \Omega \to \mathbb{R}\) be measurable functions.

  1. Show that \(f = (f_1,\dots,f_k): \Omega \to \mathbb{R}^k\) is \(\langle \mathcal{F}, \mathcal{B}(\mathbb{R}^k) \rangle\)-measurable.
  2. Show that \(g = f_1 + \dots + f_k\) is measurable.
  3. Show that the product \(h = \prod_{i=1}^k f_i\) is measurable.
  4. Show that if \(\psi: \mathbb{R}^k \to \mathbb{R}^p\) is continuous, then \(\psi \circ f\) is \(\langle \mathcal{F}, \mathcal{B}(\mathbb{R}^p) \rangle\)-measurable, where \(f = (f_1,\dots,f_k)\).

  See Proposition 2.1.3 on page 42 of A&L. As a corollary, we may say that the collection of \(\langle \mathcal{F},\mathcal{B}(\mathbb{R}) \rangle\)-measurable functions from \(\Omega\) to \(\mathbb{R}\) is closed under pointwise addition and multiplication as well as under scalar multiplication (Corollary 2.1.4 on page 42 of A&L).

• Check whether the function \[ f(x) = (1-x)1_{(0,1)}(x) - (1+x) 1_{(-1,0)}(x) \] is measurable.

• Measurability and limits: Let \((\Omega,\mathcal{F})\) be a measurable space and let \(\{f_n\}_{n\geq 1}:\Omega \to \bar{\mathbb{R}}\) be a collection of measurable functions.

  1. Show that the functions \(\inf_{n \geq 1} f_n\), \(\sup_{n \geq 1} f_n\), \(\liminf_{n \to \infty} f_n\), and \(\limsup_{n \to \infty} f_n\) are all measurable.

  2. Show that \(A \equiv \{\omega: \lim_{n \to \infty} f_n \text{ exists and is finite}\} \in \mathcal{F}\) and that the function \(h \equiv (\lim_{n \to \infty} f_n) 1_A\) is measurable.

  See Proposition 2.1.5 on page 43 of A&L, Proposition 3.2.6 on page 81 of Resnick, or Theorem 13.4 on page 184 of Billingsley.

• \(\sigma\)-algebra generated by a map: Given a measurable space \((\Omega, \mathcal{F})\) and a function \(f:\Omega \to \mathbb{R}\), define \(\sigma \langle f\rangle \equiv \sigma \langle \{f^{-1}(A): A \in \mathcal{B}(\mathbb{R})\}\rangle\). This is the \(\sigma\)-algebra generated by \(f\). It is the smallest \(\sigma\)-algebra \(\mathcal{F}\) such that \(f\) is \(\langle \mathcal{F}, \mathcal{B}(\mathbb{R}) \rangle\)-measurable.

• Find \(\sigma \langle f \rangle\) when \(f= 1_A\), \(A \subset \Omega\).

• Measure induced by a map: Let \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\) be measurable spaces and let \(T:\Omega_1\to\Omega_2\) be an \(\langle \mathcal{F}_1,\mathcal{F}_2 \rangle\)-measurable map. Moreover, let \(\mu\) be a measure on \(\mathcal{F}_1\). Show that the set function defined as \(\mu_T(A) \equiv \mu(T^{-1}(A))\) for all \(A \in \mathcal{F}_2\) is a measure on \(\mathcal{F}_2\).

  Here, \(\mu_T\) is called the measure on \(\mathcal{F}_2\) induced by the map \(T\).

  See Proposition 2.2.1 on page 45 of A&L, page 83 of Resnick.

Probability distributions

• Probability distribution of a random variable: Let \(X\) be a random variable on a probability space \((\Omega, \mathcal{F}, P)\). Then the measure \(P_X\) on \(\mathcal{B}(\mathbb{R})\) induced by \(X\) is called the probability distribution of \(X\). Show that \(P_X\) is a probability measure on \(\mathcal{B}(\mathbb{R})\), and thus that \((\mathbb{R}, \mathcal{B}(\mathbb{R}), P_X)\) is a probability space.

  One can refer to a probability distribution as a probability law or more simply as a law.

• Cumulative distribution function (CDF) of a random variable: Let \(X\) be a random variable with probability distribution \(P_X\). Then the function given by \(F_X(x) \equiv P_X((-\infty,x])\) for all \(x \in \mathbb{R}\) is called the cumulative distribution function (CDF) of \(X\).

• Properties of a CDF: Let \(X\) be a random variable on a probability space \((\Omega, \mathcal{F}, P)\). Moreover, suppose \(X\) has probability distribution \(P_X\) and let \(F_X\) be the CDF of \(X\).

  1. Show that \(F_X(x_1) \leq F_X(x_2)\) for all \(x_1 < x_2\) (non-decreasing).
  2. Show that \(F_X(x+) = F_X(x)\) for all \(x\) (right-continuous).
  3. Show that \(\lim_{x \downarrow -\infty} F_X(x)= 0\) and \(\lim_{x \uparrow +\infty} F_X(x) = 1\).

  See Proposition 2.2.2 on page 45 of A&L. Recall \(F(x+) \equiv \lim_{y \downarrow x} F_X(y)\).

• Discrete and continuous random variables: A random variable \(X\) is discrete if there exists a countable subset \(A \subset \mathbb{R}\) such that \(P_X(X \in A) = 1\) and continuous if \(P_X(X = x)= 0\) for all \(x \in \mathbb{R}\).

• Show that for a CDF \(F\) the set \(D = \{x : F(x+) - F(x-) >0\}\) of discontinuity points is finite or countable. Recall \(F(x-) \equiv \lim_{y \uparrow x}F(y)\). See Problem 2.11 on page 74 of A&L.

• Discrete and continuous components of a CDF: For a CDF \(F\) let \(D = \{x : F(x) - F(x-) >0\}\) and define the functions \[ \begin{align} \tilde F_d(x) &\equiv \sum_{\{t \in D: t\leq x\}}[F(t) - F(t-)] \\ \tilde F_c(x) &\equiv F(x) - \tilde F_d(x) \end{align} \] Then let \(\alpha \equiv \sum_{x \in D}[F(x) - F(x-)]\) and set \(F_d(x) \equiv \alpha^{-1}\tilde F_d(x)\) and \(F_c(x) \equiv (1-\alpha)^{-1}\tilde F_c(x)\). Then we may write \[ F(x) = \alpha F_d(x) + (1-\alpha) F_c(x) \] where \(F_d\) and \(F_c\) are, respectively, CDFs for a discrete and a continuous random variable. See page 47 of A&L.

• Probability distributions and CDFs in higher dimensions: Let \(X_1,\dots,X_k\) be random variables on the probability space \((\Omega,\mathcal{F},P)\). Then \(X = (X_1,\dots,X_k)\) is called a random vector. The probability distribution \(P_X\) of \(X\) is given by \[ P_X(A) \equiv P(X^{-1}(A)) = P(\{\omega: X(\omega) \in A\}), \quad A \in \mathcal{B}(\mathbb{R}). \] The CDF of \(X\) is defined as \[ F_X(x) \equiv P_X((-\infty,x_1]\times \dots \times (-\infty,x_k]) = P(\{\omega: X_1(\omega) \leq x_1,\dots, X_k(\omega) \leq x_k\}) \] for all \(x = (x_1,\dots,x_k) \in \mathbb{R}^k\).