Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

Definition 1. A {\sigma}-field {\mathcal{F}} is a non-empty collection of subsets of the sample space {\Omega} closed under the formation of complements and countable unions (or equivalently of countable intesections – note {\bigcap_{i} A_i = (\bigcup_i A_i^c)^c}). Hence {\mathcal{F}} is a {\sigma}-field if

1. {A^c \in \mathcal{F}} whenever {A \in \mathcal{F}}
2. {\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}} whenever {A_i \in \mathcal{F}, n \geq 1}

Definition 2. Set functions and measures. Let {S} be a set and {\Sigma_0} be an algebra on {S}, and let {\mu_0} be a non-negative set function

\displaystyle \mu_0: \Sigma_0 \rightarrow [0, \infty]

  • {\mu_0} is additive if {\mu_0 (\varnothing) =0} and, for {F,G \in \Sigma_0},

    \displaystyle F \cap G = \varnothing \qquad \Rightarrow \qquad \mu_0(F \cup G ) = \mu_0(F) + \mu_0(G)

  • The map {\mu_0} is called countably additive (or {\sigma}-additive) if {\mu (\varnothing)=0} and whenever {(F_n: n \in \mathbb{N})} is a sequence of disjoint sets in {\Sigma_0} with union {F = \cup F_n} in {\Sigma_0}, then

    \displaystyle \mu_0 (F) = \sum_{n}\mu_0 (F_n)

  • Let {(S, \Sigma)} be a measurable space, so that {\Sigma} is a {\sigma}-algebra on {S}.
  • A map \displaystyle \mu: \Sigma \rightarrow [0,\infty]. is called a measure on {(S, \Sigma)} if {\mu} is countable additive. The triple {(S, \Sigma, \mu)} is called a measure space.
  • The measure {\mu} is called finite if

    \displaystyle \mu(S) < \infty,

    and {\sigma}finite if

    {\exists \{S_n\} \in \Sigma}, ({n \in \mathbb{N}})\displaystyle \mu(S_n)< \infty, \forall n \in \mathbb{N} \text{ and } \cup S_n = S.

  • Measure {\mu} is called a probability measure if \displaystyle \mu(S) = 1, and {(S, \Sigma, \mu)} is then called a probability triple.
  • An element {F} of {\Sigma} is called {\mu}-null if {\mu(F)=0}.
  • A statement {\mathcal{S}} about points {s} of {\mathcal{S}} is said to hold almost everywhere (a.s.) if

    \displaystyle F \equiv \{ s: \mathcal{S}(s) \text{ is false} \} \in \Sigma \text{ and } \mu(F)=0.

Definition 3. A probability space is a triple {(\Omega, \mathcal{F}, P )} where {\Omega } is the sample space, {\mathcal{F}} is the family of events (i.e. an event is an element of {\mathcal{F}}) which we assume to be a {\sigma}-field and {P: \mathcal{F}\rightarrow [0,1]} is a function that assigns probabilities to events.

Definition 4. A non-empty class {\mathcal{C}} of subsets of {\Omega} is a monotone class if {\lim C_n \in \mathcal{C}} for every monotone serquence {C_n \in \mathcal{C}, n \geq 1}.

Proposition 1. A {\sigma}-field is a monotone class while, conversely, a monotone algebra is a {\sigma}-field.

Corollary 1. Let {2^X} be the power-set (the class of all subsets of {\Omega}) and {T_{\Omega} = \{\varnothing ,\Omega\}}.Then for any {\sigma}-field {U_{\Omega}} of subsets of {\Omega},

{ T_{\Omega} \subset U_{\Omega} \subset 2^X }.

Definition 5. A real valued function {X} defined on {\Omega} is said to be a random variable if for every Borel set {B \in \mathbb{R}} we have

\displaystyle X^{-1} (B) = \{\omega: X(\omega) \in B\} \in \mathcal{F}

A random variable is an element of {m \mathcal{F}}. Thus

\displaystyle X: \Omega \rightarrow \mathbb{R}, \qquad X^{-1} : \mathcal{B} \rightarrow \mathcal{F}.

Definition 6. A function {X: \Omega \rightarrow S} is said to be a measurable map from {(\Omega,\mathcal{F})} to {(S, \mathcal{S})} if

\displaystyle X^{-1}(B) \equiv \{ \omega: X(\omega) \in B \} \in \mathcal{F} \text{ } \forall B \in \mathcal{S}

Theorem 1. If {\{\omega: X(\omega) \in A\} \in \mathcal{F}} {\forall A \in \mathcal{A}} and { \mathcal{A} \text{ generates } \mathcal{S}}, then {X} is measurable. A {\sigma}-field generated by a class {\mathcal{C}} of subsets (denoted {\sigma(\mathcal{C})}) is the smallest {\sigma}-field on {\Omega} that makes {\mathcal{C} } a measurable map. That is { \mathcal{C} \subset \sigma(\mathcal{C}) \subset \mathcal{B'}} where {\mathcal{B'} } is any other subset containing {\mathcal{C}}.

Proposition 2. If {\{ \mathcal{F}_{i} \}_{i \in I} } is the collection of all {\sigma}-fields that contain {\mathcal{C}} then

\displaystyle \sigma(\mathcal{C}) = \bigcap_{i \in I} \mathcal{F}_i

Definition 8. The Borel {\sigma}-field {\mathcal{B}(\mathbb{R})} is the smallest {\sigma}-field containing all open intervals of {\mathbb{R}}. {\mathcal{B}(\mathbb{R})} is generated by each of the following

  • the collection of all closed subsets of {\mathbb{R}};
  • the collection of all subintervals of {\mathbb{R}} of the form {(-\infty, b]};
  • the collection of all subintervals of {\mathbb{R}} of the form {(a, b]}.

Theorem 2. If {X: (\Omega, \mathcal{F}) \rightarrow (S, \mathcal{S})} and {f: (S, \mathcal{S}) \rightarrow (T, \mathcal{T}) } are measurable maps, then {f(X)} is a measurable map from {(\Omega, \mathcal{F})} to {(T, \mathcal{T}) }.

Theorem 3. If {X_1,...,X_n} are random variables and {f:(\mathbb{R}^n, \mathcal{R}^n) \rightarrow (\mathbb{R}, \mathcal{R})} is measurable, then {f(X_1,...,X_n)} is a random variable.

Theorem 4. If {X_1, X_2...} are random variables then so are

\displaystyle \inf_n X_n \qquad \sup_n X_n \qquad \limsup_n X_n \qquad \liminf_n X_n

Definition 9. {X_n} converges almost surely (a.s.) if {P(\Omega_0) =1} where

\displaystyle \Omega_0 \equiv \{\omega : \lim\limits_{n \rightarrow \infty} X_n \text{ exists } \} = \{\omega: \limsup_{n\rightarrow\infty} X_n - \liminf_{n\rightarrow\infty} X_n = 0 \}

Definition 10. A {\pi}-system {\mathcal{I}} on a set {S} is a family of subsets of {S} stable under finite intersection:

\displaystyle I_1, I_2 \in \mathcal{I} \Rightarrow I_1 \cap I_2 \in \mathcal{I}.

Lemma 1. Let {\Sigma \equiv \sigma(\mathcal{I})}. Supposed {\mu_1} and {\mu_2} are measures on {(S,\Sigma)} {\mu_1(S) = \mu_2(S) < \infty} and {\mu_1=\mu_2} on {\mathcal{I}}. Then

\displaystyle \mu_1 = \mu_2 \text{ on } \Sigma

Corollary 1. If two probability measures agree on a {\pi}-system then they agree on the {\sigma}-algebra generated by that {\pi}-system. Example: {\mathcal{B} = \sigma(\pi(\mathbb{R}))}.

Definition 11. Dynkin Class/ {\mathcal{D}} or {\mathcal{L}} system. Let {S} be a set, and let {\mathcal{D}} be a collection of subsets of {S}. Then {\mathcal{D}} is called a d-system (on {S}) if

  1. {S \in \mathcal{D}},
  2. if {A,B \in \mathcal{D}} and {A \subseteq B} then {B-A \in \mathcal{D}}
  3. if {A_n \in \mathcal{D}} and {A_n \uparrow A}, then {A \in \mathcal{D}.}

Recall {A_n \uparrow A} means {A_n \subseteq A_{n+1}} {\forall n} and {\cup A_n =A}.

Definition 12. {d(\mathcal{C})}: The intersection of all d-systems which contain the class {\mathcal{C}} of subsets of {S}. Subsequently {d(\mathcal{C})}, is a d-system, the smallest d-system which contains {\mathcal{C}}. Furthermore

\displaystyle d(\mathcal{C}) \subseteq \sigma(\mathcal{C})

Lemma 2. Dynkin’s lemma. If {\mathcal{I}} is a {\pi}-system, then

\displaystyle d(\mathcal{I}) = \sigma(\mathcal{I}).

A d-system which contains a {\pi}-system contains the {\sigma}-algebra generated by that {\pi}-system.

Proposition 3. A collection {\Sigma} of subsets of {S } is a {\sigma}-algebra iff it is both a {\pi}-system and a d-system.

Theorem 5. Carathéodory’s Theorem. Let {S } be a set, let {\Sigma_0} be an algebra on {S}, and let

\displaystyle \Sigma \equiv \sigma( \Sigma_0)

If {\mu_0} is a countably additive map {\mu_0: \Sigma_0 \rightarrow [0, \infty], \text{ then } \exists \mu \text{ on } (S, \Sigma)}

\displaystyle \mu = \mu_0 \text{ on } \Sigma_0.

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