Measure

Paul Schrimpf

2024-09-09

References

• Song (2021) chapter 1 (which is the basis for these slides)
• Pollard (2002)
• Tao (2011)

Measures

Why Measure Theory?

1. Simplifies some arguments
   • Example from Pollard (2002), define independence as factorization of distribution functions \[ P(X \leq x \cap Y \leq y) = P(X \leq x) P(Y \leq y) \]
   • If \(X_1,X_2,X_3, X_4\) are independent, show that \[ Y = X_1 X_2 \log\left(\frac{X_1^2 + X_2^3}{|X_1| + |X_2|}\right) \] is independent of \[ Z = sin\left(X_3 + X_3^2 + X_3X_4 + X_4^2 \right) \]

\[ \def\R{\mathbb{R}} \def\B{\mathscr{B}} \]

Why Measure Theory?

1. Simplifies some arguments

2. Unifies treatment
   • discrete vs continuous
   • uni- vs multi-variate
3. Resolves some difficulties with infinity

One difficulty with infinity in probability is defining probability over uncountably infinite sets. In probability, we start with a sample space, say \(\Omega\), and then say a probability is a mapping from subsets of \(\Omega\) to \([0,1]\) satisfying some conditions. One question is can we define such a probability on all subsets of \(\Omega\). If \(\Omega\) is uncountable, then the answer no. We have to limit our attention to certain well behaved subsets. These turn out to be exactly a \(\sigma\)-field, which we now define.

Measure Space

1. A set \(\Omega\)
2. A collection of subsets, \(\mathscr{F}\), of \(\Omega\) that is a \(\sigma\)-field (aka \(\sigma\)-algebra) , that is
   1. \(\Omega \in \mathscr{F}\)
   2. If \(A \in \mathscr{F}\), then \(A^c \in \mathscr{F}\)
   3. If \(A_1, A_2, ... \in \mathscr{F}\), then \(\cup_{j=1}^\infty A_j \in \mathscr{F}\)
3. A measure, \(\mu: \mathcal{F} \to [0, \infty]\) s.t.
   1. \(\mu(\emptyset) = 0\)
   2. If \(A_1, A_2, ... \in \mathscr{F}\) are pairwise disjoint, then \(\mu\left(\cup_{j=1}^\infty A_j \right) = \sum_{j=1}^\infty \mu(A_j)\)

Example of sigma fields

Measurable Function

• Given a topology on \(\Omega\), the Borel \(\sigma\)-field, \(\mathscr{B}(\Omega)\), is the smallest \(\sigma\)-field containing all open subsets of \(\Omega\)

When working with \(R^d\), we will generally use \(\B(\R^d)\) as our \(\sigma\)-field.

• \(f: \Omega \to \R\) is (\(\mathscr{F}\)-)measurable if \(\forall\) \(B \in \B(\R)\), \(f^{-1}(B) \in \mathscr{F}\)

• a statement holds almost everywhere (a.e.) if the measure of the set where the statement is false is 0

Integration

Simple Functions

• Assume \(\mu(\Omega) < \infty\)
• \(f\) is a simple function if \(f = \sum_{j=1}^n a_j 1\{\omega \in E_j \}\) for \(a_j \in \R\) and \(E_j \in \mathscr{F}\)
• Integral of simple functions: \[ \int f d \mu = \sum_{j=1}^n a_j \mu(E_j) \]

Exercise

Show that if \(f\) is simple, then \(f\) is measurable.

Exercise

Show that if \(f\) and \(g\) are simple, then

1. If \(f \geq 0\) a.e., then \(\int f d\mu \geq 0\)

2. \(\forall a \in \R\) , \(\int a f d\mu = a \int f d\mu\)

3. \(\int (f + g) d\mu = \int f d\mu + \int g d \mu\)

Bounded Functions

• Let \(E\) be such that \(\mu(E)<\infty\)
• Let \(f\) be bounded function and \(f(x) = 0 \forall x \in E^c\)
• Define: \[ \int f d\mu \equiv \sup_{\varphi \leq f: \varphi \text{ simple}} \int \varphi d\mu =\inf_{\varphi \geq f: \varphi \text{ simple}} \int \varphi d\mu \]

Exercise

Show the second equality above.

Exercise

Show that for all bounded \(f\) and \(g\) that vanish outside a finite measure set,

1. If \(f \geq 0\) a.e., then \(\int f d\mu \geq 0\)

2. \(\forall a \in \R\) , \(\int a f d\mu = a \int f d\mu\)

3. \(\int (f + g) d\mu = \int f d\mu + \int g d \mu\)

Nonnegative Functions

• If \(f \geq 0\), define \[ \int fd\mu =\sup_{f_n \leq f \text{ simple, bounded+}} \int f_{n}d\mu \]

Exercise

Show that for all \(f \geq 0\),

1. If \(f \geq 0\) a.e., then \(\int f d\mu \geq 0\)

2. \(\forall a \in \R\) , \(\int a f d\mu = a \int f d\mu\)

3. \(\int (f + g) d\mu = \int f d\mu + \int g d \mu\)

Measurable Functions

• If \(f\) is measurable, let \(f^{+} = \max\{f, 0\}\) and \(f^{-} = \max\{-f, 0\}\) and define the Lesbegue integral

\[ \int f d\mu = \int f^{+} d\mu - \int f^{-} d\mu \]

Exercise

Show that for all measurable \(f\)

1. If \(f \geq 0\) a.e., then \(\int f d\mu \geq 0\)

2. \(\forall a \in \R\) , \(\int a f d\mu = a \int f d\mu\)

3. \(\int (f + g) d\mu = \int f d\mu + \int g d \mu\)

Measurability and Continuity

Lemma

If \(f: \R^n \to \R^k\) is continuous, then \(f\) is \(\B(\R^n)\) measurable.

Corollary

If \(f: \Omega \to \R\) is measurable and \(g: \R \to \R\) is continuous, then \(g \circ f\) is measurable.

Corollary

If \(f: \Omega \to \R\) and \(g: \Omega \to \R\) are measurable, then \(f+g\) is measurable.

Radon-Nikodym Derivatives

Finite Measure

• Measure \(\mu\) is finite if \(\mu(\Omega)\) is finite
• \(\mu\) is \(\sigma\)-finite if \(\exists\) \(\{A_n\}_{n=1}^\infty \in \mathscr{F}\) s.t. \(\mu(A_n)\) is finite \(\forall n\) and \(\cup_{n=1}^\infty A_n = \Omega\)

Exercise

Let \(\Omega\) be countable with any \(\mathscr{F}\), define \(\mu(A)\) as the number of elements of \(A\). Show \(\mu\) is \(\sigma\) finite.

Lebesgue Measure

Theorem

There exists a unique \(\sigma\)-finite measure \(\mu\) on \((\mathbf{R},\mathscr{B}(\mathbf{R}))\) such that for any \(a\leq b\) with \(a,b\in \mathbf{R}\), \[ \mu ((a,b])=b-a \]

Absolute Continuity

• Measure \(\nu\) is absolutely continuous with respect to \(\mu\) if for \(A \in \mathscr{F}\), \(\mu(A) = 0\) implies \(\nu(A) = 0\)
  • denotate as \(\nu \ll \mu\)
  • \(\mu\) is called a dominating measure

Relation with continuity?

Radon-Nikodym Derivative

Theorem

Let \((\Omega,\mathscr{F},\mu)\) be a measure space, and let \(\nu\) and \(\mu\) be \(\sigma\)-finite measures defined on \(\mathscr{F}\) and \(\nu \ll \mu\). Then there is a nonnegative measurable function \(f\) such that for each set \(A\in \mathscr{F}\), \[ \nu (A)=\int_{A}fd\mu \] For any such \(f\) and \(g\), \(\mu (\{\omega \in \Omega:f(\omega )\neq g(\omega )\})=0\)

Denote $f =

Exercise: show coincides with usual derivative?

Convergence Theorems

Sequences of Sets

• \(\{E_n\}_{n \geq 1} \in \mathscr{F}\)
  • increasing if \(E_1 \subset E_2 \subset ...\)
  • decreasing if \(E_1 \supset E_2 \supset ...\)
  • monotone if either increasing or decreasing
• For increasing \(E_n\), define \(\lim_{n \to \infty} E_n =\cup_{n=1}^\infty E_n\)
• For decreasing \(E_n\), define \(\lim_{n \to \infty} E_n =\cap_{n=1}^\infty E_n\)

Continuity of Measure

Lemma

Suppose that \(\{E_{n}\}\) is a monotone sequence of events. Then \[ \mu \left( \lim_{n\rightarrow \infty}E_{n}\right) =\lim_{n\rightarrow \infty }\mu (E_{n}). \]

Monotone Convergence Theorem

Lemma

If \(f_n:\Omega \to \mathbf{R}\) are measurable, \(f_{n}\geq 0\), and for each \(\omega \in \Omega\), \(f_{n}(\omega )\uparrow f(\omega )\), then \(\int f_{n}d\mu \uparrow \int fd\mu\) as \(n\rightarrow \infty\)

Exercise

Suppose \(g_n: \Omega \to \R\) are measurable and \(g_n \geq 0\). Show that \[ \int \sum_{n=1}^\infty g_n d\mu = \sum_{n=1}^\infty \int g_n d\mu \]

How do we know \(f = \lim_{n\to \infty} f_n\) is measurable?

Note that half open intervals \((-\infty, c)\) generate \(\B(\R)\), so it suffices to show that \(f^{-1}((-\infty, c))\) is in \(\mathscr{F}\). Since \(f_n\) are measurable, \(f_n^{-1}((-\infty,c)) \in \mathscr{F}\) for all \(n\). Also, \[ f_n^{-1}((-\infty,c)) = \{x: f_n(x) < c\} \] and since \(f_n\) are increasing, \(f_n^{-1}((-\infty,c)) \supset f_m^{-1}((-\infty, c))\) for an \(m \geq n\). Moreover, \[ f^{-1}((-\infty,c)) = \{x : f(x) < c \} = \cup_{n=1}^\infty \{x: f_n(x) < c\} \] and hence is in \(\mathscr{F}\).

Fatou’s Lemma

Lemma

If \(f_n:\Omega \to \R\) are measurable, \(f_{n}\geq 0\), then \[ \int \left( \text{liminf}_{n\rightarrow \infty }f_{n}d\mu \right) \leq \text{liminf}_{n\rightarrow \infty }\int f_{n}d\mu \]

Dominated Convergence Theorem

Lemma

If \(f_n:\Omega \to \R\) are measurable, and for each \(\omega \in \Omega\), \(f_{n}(\omega )\rightarrow f(\omega ).\) Furthermore, for some \(g\geq 0\) such that \(\int gd\mu <\infty\), \(|f_{n}|\leq g\) for each \(n\geq 1\). Then, \(\int f_{n}d\mu \rightarrow \int fd\mu\)

Exercise

Show that if \(g_n: \Omega \to \R\) are measurable and \(\sum_{n=1}^\infty \int |g_n| d\mu < \infty\), then \[ \int \sum_{n=1}^\infty g_n d\mu = \sum_{n=1}^\infty \int g_n d\mu \]

The measurable condition here can be dropped and outermeasure used instead. This simplifies some proofs in asymptotic theory.

References

Pollard, David. 2002. A User’s Guide to Measure Theoretic Probability. 8. Cambridge University Press.

Song, Kyunchul. 2021. “Introduction to Econometrics.”

Tao, Terence. 2011. An Introduction to Measure Theory. Vol. 126. American Mathematical Society Providence.