IV. Integration and convergence

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

Integration

• Simple function: Given a measurable space \((\Omega,\mathcal{F})\), a function \(f:\Omega \to \bar{\mathbb{R}} \equiv [-\infty,\infty]\) is called a simple function if it has the form \(f = \sum_{i=1}^k c_i 1_{A_i}\), where \(c_1,\dots,c_k \in [0,\infty]\) are distinct values and \(A_1,\dots,A_k \in \mathcal{F}\) are distinct subsets of \(\Omega\), and \(k\) is finite.

• Check: Are simple functions measurable?

• Integral of a simple function: Given a measure space \((\Omega, \mathcal{F}, \mu)\), the integral with respect to \(\mu\) of a simple nonnegative function \(f:\Omega \to [0,\infty]\) with representation \(f = \sum_{i=1}^k c_i 1_{A_i}\) is defined as \[ \int f d\mu \equiv \sum_{i=1}^k c_i \mu (A_i), \] where we use the convention \(0 \cdot \infty = 0\). See Definition 2.3.2 on page 49 of A&L. Compare to Section 5.2.1 starting on page 119 of Resnick. Can show that the integral is invariant to the representation of \(f\); see Problem 2.17 on Page 76 of A&L.

• Results on integrals of simple functions: Let \(f,g: \Omega \to \mathbb{R}\) be nonnegative simple functions on a measure space \((\Omega,\mathcal{F},\mu)\).

  1. Show \(\int (a f + b g) d \mu = a \int f d \mu + b \int g d \mu\) for any \(a,b \in \mathbb{R}\) (Linearity).
  2. Show that \(f \geq g\) almost everywhere implies \(\int f d\mu \geq \int g d \mu\) (Monotonicity).
  3. Show that \(f = g\) almost everywhere implies \(\int f d \mu = \int g d \mu\).

  Note: If a condition holds almost everywhere (a.e.) it means that the subset of \(\omega \in \Omega\) for which it does not hold has measure zero. For example, \(f \geq g\) almost everywhere means \(\mu(\{\omega : f(\omega) < g(\omega)\}) = 0\).

• Integral of a nonnegative function: Let \(f:\Omega \to [0,\infty]\) be a nonnegative measurable functions on the measure space \((\Omega,\mathcal{F},\mu)\) and let \(\{f_n\}_{n\geq 1}:\Omega \to [0,\infty]\) be a sequence of simple nonnegative functions on the same measure space such that \(f_n(\omega) \uparrow f(\omega)\) as \(n \to \infty\) for all \(\omega \in \Omega\). Then the integral of \(f\) with respect to \(\mu\) is defined as \[ \int f d \mu \equiv \lim_{n \to \infty} \int f_n d\mu. \]

• Invariance of integral to the sequence of simple functions: In the setup of the previous definition, let \(\{f_n\}_{n \geq 1}\) and \(\{g_n\}_{n \geq 1}\) be sequences of simple nonnegative functions such that \(f_n(\omega) \uparrow f(\omega)\) and \(g_n(\omega) \uparrow f(\omega)\) as \(n \to \infty\) for all \(\omega \in \Omega\). Show that \[ \lim_{n \to \infty} \int f_n d\mu = \lim_{n \to \infty} \int g_n d\mu. \]

• Note that we really only need \(f_n \uparrow f\) and \(g_n \uparrow f\) as \(n \to \infty\) almost everywhere.

• Example of a sequence of simple nonnegative functions: On the measure space \((\mathbb{R}, \mathcal{B}(\mathbb{R}),\mu_L)\), say, where \(\mu_L\) is the Lebesgue measure, one can construct a sequence of simple functions \(f_n \uparrow f\) as \[ f_n(x) = \left\{\begin{array}{ll} j(1/2)^n,& j(1/2)^n \leq f(x) < (j+1)(1/2)^n, \quad j = 0,1,\dots,n2^n - 1\\ n,& f(x) \geq n \end{array} \right. \] Draw a picture of this sequence of functions for \(n =1,2\) for some nonnegative function \(f:\mathbb{R}\to [0,\infty]\). Note that subdividing the domain as in Riemann integration is not guaranteed to result in a sequence of functions approaching \(f\) from below. See page 51 of A&L.

• Monotone convergence theorem (MCT): Let \(\{f_n\}_{n \geq 1}\) and \(f\) be nonnegative measurable functions on a measure space \((\Omega,\mathcal{F},\mu)\) such that \(f_n \uparrow f\) as \(n \to \infty\) almost everywhere. Show that \[ \lim_{n \to \infty} \int f_n d \mu = \int \lim_{n \to \infty} f_n d\mu. \] See Corollary 2.3.5 on page 53 of A&L.

• Integral of infinite sum of nonnegative functions: Let \(\{f_n\}_{n \geq 1}\) be nonnegative measurable functions on a measure space \((\Omega, \mathcal{F},\mu)\). Show (use the MCT) that \[ \int \sum_{n \geq 1} f_n d \mu = \sum_{n\geq 1} \int f_n d \mu. \] See Corollary 2.3.5 on page 53 of A&L.

• Fatou’s Lemma: Let \(\{f_n\}_{n \geq 1}\) be nonnegative measurable functions on a measure space \((\Omega,\mathcal{F},\mu)\). Show that \[ \liminf_{n \to \infty} \int f_n d \mu \geq \int \liminf_{n \to \infty}f_n d \mu. \]

  See Theorem 2.3.7 on page 54 of A&L. Compare to Theorem 5.3.2 on page 132 of Resnick.

• Definition of the integral of a measurable function (not necessarily a nonnegative function).

• Integral of any measurable function: Let \(f\) be a measurable function (not necessarily nonnegative) on a measure space \((\Omega,\mathcal{F},\mu)\). Define \(A \equiv \{\omega : f(\omega) \geq 0\}\) as well as the nonnegative functions \(f^+ \equiv f 1_A\) and \(f^- \equiv -f 1_{A^c}\). Then the integral of \(f\) with respect to \(\mu\) is defined as \[ \int f d \mu \equiv \int f^+ d \mu - \int f^- d \mu. \]

• Integral over a subset of the sample space: For a function \(f\) on a measure space \((\Omega,\mathcal{F},\mu)\) and a set \(A \in \mathcal{F}\), the integral of \(f\) over \(A\) with respect to \(\mu\) is defined as \[ \int_A f d \mu \equiv \int f 1_A d \mu. \]

• Make a new measure by integrating a nonnegative function: Let \(f\) be a nonnegative measurable function on a measure space \((\Omega,\mathcal{F},\mu)\). Show that the set function defined as \[ \nu(A) \equiv \int f 1_A d\mu \] for all \(A\in \mathcal{F}\) is a measure on \(\mathcal{F}\). See Corollary 2.3.6 on page 53 of A&L.

• Integrability of a function: A function \(f\) on a measure space \((\Omega,\mathcal{F},\mu)\) is integrable with respect to \(\mu\) if \(\int |f|d \mu\) is finite.

• Notation for integration: The integral of \(f\) with respect to \(\mu\) can be written as \[ \int f d \mu \quad \text{ or } \quad \int f(\omega) d \mu(\omega) \quad \text{ or } \quad \int f(\omega) \mu(d \omega). \]

• \(L_p\) spaces: Given a measure space \((\Omega,\mathcal{F},\mu)\), define the collections of functions \[ \begin{align} L^p(\Omega,\mathcal{F},\mu) &\equiv \{f: \int |f|^p d \mu < \infty\}, \quad p \in (0,\infty)\\ L^\infty(\Omega,\mathcal{F},\mu) &\equiv \{f: \mu(\{\omega : f(\omega) > C \}) = 0 \text{ for some } C \in (0,\infty)\}. \end{align} \]

• \(L_p\) norms: Given a function \(g\) on a measure space \((\Omega,\mathcal{F},\mu)\), define \[ \|g\|_p = \left\{\begin{array}{ll} \int |g|^p d \mu& 0 < p < 1\\ (\int |g|^p d \mu)^{1/p},& 1 \leq p<\infty \\ \inf\{c \in (0,\infty) : \mu(\{\omega : g(\omega) > c\}) = 0\},& p = \infty \end{array}\right. \]

• Results on integrals of integrable functions: Let \(f,g \in L^1(\Omega,\mathcal{F},\mu)\).

  1. Show \(\int (a f + b g) d \mu = a \int f d \mu + b \int g d \mu\) for any \(a,b \in \mathbb{R}\) (Linearity).
  2. Show that \(f \geq g\) almost everywhere implies \(\int f d\mu \geq \int g d \mu\) (Monotonicity).
  3. Show that \(f = g\) almost everywhere implies \(\int f d \mu = \int g d \mu\).

• Identification of zero function: Let \(f\) be a measurable function on a measure space \((\Omega,\mathcal{F},\mu)\) which is nonnegative almost everywhere. Show that \(f = 0\) almost everywhere if and only if \(\int f d \mu = 0\). See Proposition 2.3.9 on page 56 of A&L.

• Finiteness almost everywhere of an integrable function: Let \(f\in L^1(\Omega,\mathcal{F},\mu)\). Show that \(f\) is finite almost everywhere. See Proposition 2.3.10 on page 57 of A&L.

• Two modes of convergence: Let \(\{f_n\}_{n \geq 1}\) and \(f\) be measurable functions on the measurable space \((\Omega,\mathcal{F},\mu)\). We say \(f_n\) converges to \(f\)

  • pointwise if \(\lim_{n \to \infty} f_n(\omega) = f(\omega)\) for all \(\omega \in \Omega\). We write \(f_n \to f\).

  • almost everywhere if there is a set \(B \in \mathcal{F}\) such that \(\lim_{n \to \infty} f_n(\omega) = f(\omega)\) for all \(\omega \in B\) and \(\mu(B^c)=0\). We write \(f_n \stackrel{\operatorname{a.e.}}{\longrightarrow}f\).

• Dominated convergence theorem (DCT): Let \(\{f_n\}_{n \geq 1}\), \(f\), and \(g\) be measurable functions on a measurable space \((\Omega,\mathcal{F},\mu)\) such that \(|f_n| < g\) almost everywhere for all \(n \geq 1\), \(g\) is integrable, and \(f_n \to f\) almost everywhere.

  1. Show that \(f\) is integrable.
  2. Show that \(\lim_{n\to \infty} \int f_nd \mu = \int f d \mu\).
  3. Show that \(\lim_{n \to \infty}\int |f_n - f| d\mu = 0\)

  See Corollary 2.3.12 on page 57 of A&L or Theorem 16.4 on page 209 of Billingsley. Compare to Theorem 5.3.3 on page 133 of Resnick.

Convergence

• More modes of convergence: Let \(\{f_n\}_{n \geq 1}\) and \(f\) be measurable functions on a measure space \((\Omega,\mathcal{F},\mu)\). We say \(f_n\) converges to \(f\)

  • in measure if \(\lim_{n \to \infty} \mu(\{\omega: |f_n(\omega) - f(\omega)| > \epsilon\}) = 0\) for each \(\epsilon > 0\). We write \(f_n \stackrel{\mu}{\longrightarrow}f\).

  • in \(L^p\) if \(\lim_{n\to \infty} \|f_n - f\|_p = 0\). We write \(f_n \stackrel{L_p}{\longrightarrow}f\).

  • uniformly over \(~\Omega\) if \(\lim_{n \to \infty} \sup_{\omega \in \Omega} |f_n(\omega) - f(\omega)| = 0\).

• Some relationships between modes of convergence: Let \(\{f_n\}_{n \geq 1}\) and \(f\) be measurable functions on a measure space \((\Omega,\mathcal{F},\mu)\).

  1. Show that \(f_n \stackrel{\operatorname{a.e.}}{\longrightarrow}f\) implies \(f_n \stackrel{\mu}{\longrightarrow}f\).

  2. Show that \(f_n \stackrel{\mu}{\longrightarrow}f\) implies that there exists a subsequence \(\{n_k\}_{k \geq 1}\) such that \(f_{n_k} \stackrel{\operatorname{a.e.}}{\longrightarrow}f\).

  3. Show that \(f_n \stackrel{L_p}{\longrightarrow}f\) for \(0<p<\infty\) implies \(f_n \stackrel{\mu}{\longrightarrow}f\).

  See Theorems 2.5.1, 2.5.2, and 2.5.3 on page 63 of A&L. Compare to Theorems 6.2.1 and 6.3.1 on pages 170 and 171 of Resnick. Note to myself: See Page 31-32 of handwritten notes.

• Scheffé’s theorem: Let \(\{f_n\}_{n \geq 1}\) and \(f\) be measurable functions on a measurable space \((\Omega,\mathcal{F},\mu)\) such that \(f_n \stackrel{\operatorname{a.e.}}{\longrightarrow}f\), \(\int f_n d \mu \to \int f d \mu\), and \(\int f d \mu < \infty\). Show that \(f_n\stackrel{L_1}{\rightarrow} f\).

  See Theorem 2.5.4 on page 64 of A&L.