Measure

Paul Schrimpf

2023-09-11

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 \[ Z = sin\left(X_3 + X_3^2 + X_3X_4 + X_4^2 \right) \]

Why Measure Theory?

1. Simplifies some arguments

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

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 \(\mathscr{B}(\mathbf{R}^d)\) as our \(\sigma\)-field.

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

• 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 \(a_j \in \mathbf{R}\) 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. \(\forall a \in \mathbf{R}\) , \(\int a f d\mu = a \int f d\mu\)

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. \(\forall a \in \mathbf{R}\) , \(\int a f d\mu = a \int f d\mu\)

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. \(\forall a \in \mathbf{R}\) , \(\int a f d\mu = a \int f d\mu\)

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. \(\forall a \in \mathbf{R}\) , \(\int a f d\mu = a \int f d\mu\)

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

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 \(n\rightarrow \infty\)

Fatou’s Lemma

Lemma

If \(f_n:\Omega \to \mathbf{R}\) 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 \(f_n:\Omega \to \mathbf{R}\) 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, \(\int f_{n}d\mu \rightarrow \int fd\mu\)

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.