Describe the process that extends a measure on an algebra \(\mathcal{A}\) of subsets of \(X\), to a complete measure defined on a \(\sigma\)-algebra \(\mathcal{B}\) containing \(\mathcal{A}\). State the corresponding definitions and results (without proofs).
Solution
1. Definitions.
1.1. Algebra: A collection \(\mathcal A \subseteq \mathcal P(X)\) of subsets of \(X\) is an algebra on \(X\) if:1. \(\emptyset, X \in \mathcal A\);2. \(A \in \mathcal A \implies A^c = X \setminus A \in \mathcal A\);3. \(A, B \in \mathcal A \implies A \cup B \in \mathcal A\) (closed under finite unions).
1.2. Pre-measure (Measure on an Algebra): A function \(\mu_0: \mathcal A \to [0, \infty]\) is a pre-measure on \(\mathcal A\) if:1. \(\mu_0(\emptyset) = 0\);2. Countable additivity on \(\mathcal A\): if \(\{A_n\}_{n=1}^\infty \subseteq \mathcal A\) is a pairwise disjoint sequence such that \(\bigcup_{n=1}^\infty A_n \in \mathcal A\), then\[ \mu_0\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu_0(A_n). \]
1.3. Induced Outer Measure: The outer measure \(\mu^*: \mathcal P(X) \to [0, \infty]\) induced by \(\mu_0\) is defined for any \(E \subseteq X\) by:\[ \mu^*(E) \definedas \inf \left\{ \sum_{n=1}^\infty \mu_0(A_n) : A_n \in \mathcal A \text{ for each } n \in \NN, \text{ and } E \subseteq \bigcup_{n=1}^\infty A_n \right\}, \]with the convention \(\inf \emptyset = \infty\).
1.4. Carathéodory Measurability: A subset \(E \subseteq X\) is called \(\mu^*\)-measurable (or Carathéodory-measurable) if for every “test set” \(T \subseteq X\),\[ \mu^*(T) = \mu^*(T \cap E) + \mu^*(T \cap E^c). \](Since \(\mu^*(T) \leq \mu^*(T \cap E) + \mu^*(T \cap E^c)\) always holds by subadditivity, measurability is equivalent to \(\mu^*(T) \geq \mu^*(T \cap E) + \mu^*(T \cap E^c)\) for all \(T\) with \(\mu^*(T) < \infty\).)We denote the collection of all \(\mu^*\)-measurable sets by \(\mathcal M(\mu^*)\) (or \(\mathcal B\)).
2. Properties and Intermediate Results of the Construction.
2.1. Properties of \(\mu^*\):1. \(\mu^*(\emptyset) = 0\);2. Monotonicity: \(E_1 \subseteq E_2 \implies \mu^*(E_1) \leq \mu^*(E_2)\);3. Countable subadditivity: \(\mu^*\left(\bigcup_{n=1}^\infty E_n\right) \leq \sum_{n=1}^\infty \mu^*(E_n)\) for any sequence \(\{E_n\}_{n=1}^\infty \subseteq \mathcal P(X)\).
2.2. Consistency on the Base Algebra:For every \(A \in \mathcal A\), \(\mu^*(A) = \mu_0(A)\), so \(\mu^*\) agrees with \(\mu_0\) on \(\mathcal A\).
3. The Extension Theorems.
3.1. Carathéodory’s Theorem:Let \(\mu^*\) be an outer measure on \(X\), and let \(\mathcal M(\mu^*)\) be the collection of all \(\mu^*\)-measurable sets. Then:1. \(\mathcal M(\mu^*)\) is a \(\sigma\)-algebra containing all sets of \(\mu^*\)-outer measure zero (i.e. \(\mu^*(N) = 0 \implies N \in \mathcal M(\mu^*)\));2. The restriction \(\mu \definedas \mu^*|_{\mathcal M(\mu^*)}\) is a countably additive measure on \(\mathcal M(\mu^*)\);3. The measure space \((X, \mathcal M(\mu^*), \mu)\) is complete (every subset of a measure-zero set belongs to \(\mathcal M(\mu^*)\) and has measure zero).
3.2. Inclusion of the Algebra:If \(\mu^*\) is induced by a pre-measure \(\mu_0\) on an algebra \(\mathcal A\), then \(\mathcal A \subseteq \mathcal M(\mu^*)\). Consequently, \(\sigma(\mathcal A) \subseteq \mathcal M(\mu^*)\), where \(\sigma(\mathcal A)\) is the \(\sigma\)-algebra generated by \(\mathcal A\).
3.3. Hahn-Kolmogorov (Carathéodory) Extension and Uniqueness Theorem:Let \(\mu_0\) be a pre-measure on an algebra \(\mathcal A\), and let \(\mu^*\) and \(\mathcal M(\mu^*)\) be constructed as above.1. Existence: \(\mu = \mu^*|_{\mathcal M(\mu^*)}\) is a complete measure on \(\mathcal M(\mu^*)\) extending \(\mu_0\) (meaning \(\mu(A) = \mu_0(A)\) for all \(A \in \mathcal A\)).2. Uniqueness: If \(\mu_0\) is \(\sigma\)-finite on \(\mathcal A\) (i.e., there exists a sequence \(\{A_n\}_{n=1}^\infty \subseteq \mathcal A\) such that \(X = \bigcup_{n=1}^\infty A_n\) and \(\mu_0(A_n) < \infty\) for all \(n\)), then the extension \(\mu|_{\sigma(\mathcal A)}\) to the generated \(\sigma\)-algebra \(\sigma(\mathcal A)\) is unique. Furthermore, \(\mathcal M(\mu^*)\) is the completion of \(\sigma(\mathcal A)\) with respect to \(\mu|_{\sigma(\mathcal A)}\).
4. Summary of the Process.1. Start with \((X, \mathcal A, \mu_0)\).2. Construct the outer measure \(\mu^*\) on \(\mathcal P(X)\) via infima of countable covers by elements of \(\mathcal A\).3. Identify the \(\sigma\)-algebra \(\mathcal B = \mathcal M(\mu^*)\) of sets satisfying Carathéodory’s criterion.4. Restrict \(\mu^*\) to \(\mathcal B\) to obtain the complete measure space \((X, \mathcal B, \mu)\), which contains \(\sigma(\mathcal A)\) and extends \(\mu_0\). Q.E.D.