Let's go through each question step by step: --- ### **Question 1: Counting Measure** Let \( X \) be a non-empty set. Let \( P(X) \) denote the power set of \( X \). The **counting measure** \( c \) is defined by: - \( c(A) = n \) if \( A \subset X \) and \( A \) has \( n \) elements, - \( c(A) = \infty \) if \( A \) is infinite. #### **(a) Show that \( c \) is a measure on \( (X, P(X)) \)** **Recall**: A measure \( \mu \) on \( (X, \mathcal{A}) \) satisfies: 1. \( \mu(\emptyset) = 0 \) 2. \( \mu \) is countably additive: For all disjoint \( (A_i)_{i=1}^\infty \subset \mathcal{A} \), \( \mu\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \mu(A_i) \) **Step 1:** \( c(\emptyset) = 0 \) (since the empty set has 0 elements). **Step 2: Countable Additivity** Let \( (A_i)_{i=1}^\infty \) be a sequence of pairwise disjoint subsets of \( X \). - If at least one \( A_i \) is infinite, then \( \bigcup_{i=1}^\infty A_i \) is infinite, so \( c\left(\bigcup_{i=1}^\infty A_i\right) = \infty = \sum_{i=1}^\infty c(A_i) \). - If all \( A_i \) are finite, then \( \bigcup_{i=1}^\infty A_i \) is at most countable, and \( c\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \#A_i = \sum_{i=1}^\infty c(A_i) \). **Therefore**: \( c \) is a measure on \( (X, P(X)) \). --- #### **(b) Show that \( (X, P(X), c) \) is a complete measure space** **Recall**: A measure space is **complete** if every subset of every null set is measurable and has measure zero. - Null set: \( A \subset X \) with \( c(A) = 0 \implies A = \emptyset \). - Any subset of \( \emptyset \) is \( \emptyset \), which is in \( P(X) \) and has measure zero. **Therefore**: \( (X, P(X), c) \) is complete. --- #### **(c) Show that if \( X \) has the discrete topology, \( c \) is a Borel measure on \( X \)** - The Borel \( \sigma \)-algebra on the discrete topology is \( P(X) \), since every subset is open. - Since \( c \) is defined on \( P(X) \), and the Borel sets are \( P(X) \), \( c \) is a Borel measure. --- ### **Question 3: Dirac Measure** Let \( (X, \mathcal{M}) \) be a measurable space. Fix \( x \in X \). Define \( \delta_x(A) = 1 \) if \( x \in A \), \( 0 \) otherwise. #### **(a) Show that \( \delta_x \) is a measure** - \( \delta_x(\emptyset) = 0 \). - For disjoint \( (A_i)_{i=1}^\infty \): - If \( x \notin \bigcup A_i \), then \( \delta_x(\bigcup A_i) = 0 \), and \( \sum \delta_x(A_i) = 0 \). - If \( x \in A_{i_0} \) for some \( i_0 \), then \( x \notin A_i \) for \( i \neq i_0 \) (since the \( A_i \) are disjoint), so \( \delta_x(A_{i_0}) = 1 \), rest are 0. - So \( \delta_x(\bigcup A_i) = 1 = \sum \delta_x(A_i) \). **Therefore**: \( \delta_x \) is a measure. --- #### **(b) If \( X \) is a topological space and \( \mathcal{M} = \mathcal{B}(X) \) is the Borel \( \sigma \)-algebra, show \( \delta_x \) is a Borel measure** - \( \delta_x \) is defined on all Borel sets, so it is a Borel measure. --- ### **Question 4: Completing a Measure Space** Let \( (X, \mathcal{F}, \mu) \) be a measure space. #### **(i) Define** \[ \mathcal{F}_0 = \{A \cup B : A \in \mathcal{F}, B \subset C \text{ for some } C \in \mathcal{F}, \mu(C) = 0\} \] **Show that \( \mathcal{F}_0 \) is a \( \sigma \)-algebra:** - \( \emptyset = \emptyset \cup \emptyset \in \mathcal{F}_0 \). - Closed under complements: - \( A \cup B \in \mathcal{F}_0 \), \( B \subset C \), \( \mu(C) = 0 \). - \( (A \cup B)^c = A^c \cap B^c = (A^c \cap C^c) \cup (A^c \cap (C \setminus B)) \). - Both \( A^c \cap C^c \) is in \( \mathcal{F} \) (since \( \mathcal{F} \) is algebra), and \( A^c \cap (C \setminus B) \subset C \) with \( \mu(C) = 0 \). - So \( (A \cup B)^c \in \mathcal{F}_0 \). - Closed under countable unions: - Each \( A_n \cup B_n \) with \( B_n \subset C_n \), \( \mu(C_n) = 0 \). - \( \bigcup (A_n \cup B_n) = (\bigcup A_n) \cup (\bigcup B_n) \). - \( \bigcup A_n \in \mathcal{F} \), \( \bigcup B_n \subset \bigcup C_n \) (a null set). - So \( \bigcup (A_n \cup B_n) \in \mathcal{F}_0 \). --- #### **(ii) Define \( \mu_0(A \cup B) = \mu(A) \), show \( \mu_0 \) is a measure on \( \mathcal{F}_0 \)** - Well-defined: If \( A \cup B = A' \cup B' \), then \( \mu(A) = \mu(A') \) (since \( B, B' \) are null). - \( \mu_0(\emptyset) = \mu(\emptyset) = 0 \). - Countable additivity: For disjoint \( E_n = A_n \cup B_n \), \[ \mu_0\left(\bigcup E_n\right) = \mu\left(\bigcup A_n\right) = \sum \mu(A_n) = \sum \mu_0(E_n) \] (since the null sets contribute nothing). --- #### **(iii) \( (X, \mathcal{F}_0, \mu_0) \) is complete** - Any subset of a null set \( N \in \mathcal{F}_0 \) is in \( \mathcal{F}_0 \) and has measure zero. --- ### **Question 5: The Extended Real Line** #### **(i) Show that the definition gives a topology on \( \overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, \infty\} \)** - The collection \( \mathcal{T} \) consists of sets \( U \subset \overline{\mathbb{R}} \) such that: - \( U \cap \mathbb{R} \) is open in \( \mathbb{R} \). - If \( \infty \in U \), \( \exists n \) s.t. \( (n, \infty] \subset U \). - If \( -\infty \in U \), \( \exists n \) s.t. \( [-\infty, n) \subset U \). **Check topology axioms:** - \( \emptyset, \overline{\mathbb{R}} \) are open. - Arbitrary union: Each condition is preserved under unions. - Finite intersection: Each condition is preserved under intersections. --- #### **(ii) Show that the Borel \( \sigma \)-algebra \( \mathcal{B}(\overline{\mathbb{R}}) = \{E \subset \overline{\mathbb{R}} : E \cap \mathbb{R} \in \mathcal{B}\} \)** - Every open set in \( \overline{\mathbb{R}} \) has the form \( V \cup S \), with \( V \) open in \( \mathbb{R} \), and possibly containing \( \infty \), \( -\infty \) as above. - The Borel \( \sigma \)-algebra on \( \overline{\mathbb{R}} \) is generated by these open sets. - Every Borel set in \( \overline{\mathbb{R}} \) is determined by its intersection with \( \mathbb{R} \), plus possible inclusion of \( \infty \), \( -\infty \). - Conversely, every \( E \subset \overline{\mathbb{R}} \) with \( E \cap \mathbb{R} \in \mathcal{B} \) is in \( \mathcal{B}(\overline{\mathbb{R}}) \). --- ### **Summary Table** | Property | Key Step | |-----------------------------------|----------------------------------------------------------------------------------------------------------------| | Counting measure is a measure | Check \( c(\emptyset) = 0 \), countable additivity, all subsets measurable | | Counting measure is complete | Only null set is \( \emptyset \), all subsets are measurable | | Counting measure is Borel measure | Discrete topology: all subsets are Borel, so \( c \) is Borel measure | | Dirac measure is a measure | \( \delta_x(\emptyset) = 0 \), countable additivity by cases | | Dirac is Borel measure | Defined on all Borel sets | | \( \mathcal{F}_0 \) is \( \sigma \)-alg | Closure under complements and countable unions | | \( \mu_0 \) is measure | Well-defined, countable additivity | | \( (\mathcal{F}_0, \mu_0) \) complete | Subsets of null sets are in \( \mathcal{F}_0 \), measure zero | | Extended real line topology | Satisfies open set axioms | | Borel sets on \( \overline{\mathbb{R}} \) | \( \mathcal{B}(\overline{\mathbb{R}}) = \{E \subset \overline{\mathbb{R}}: E \cap \mathbb{R} \in \mathcal{B}\} \) | If you want any part in even more detail, let me know!