Question

2. Completion ?-Algebras Let $(X, \Sigma, \mu)$ be a measure space and $(X, \bar{\Sigma}, \bar{\mu})$ be its completion measure space. Prove that $\bar{\Sigma} = \{A \triangle B: A \in \Sigma, B \in \mathcal{N}\}$. Recall that $\mathcal{N} = \{B \subset X: B \subset N, N \in \Sigma, \mu(N) = 0\}$ and $\bar{\Sigma} = \{A \cup B: A \in \Sigma, B \in \mathcal{N}\}$.

          2. Completion ?-Algebras
Let $(X, \Sigma, \mu)$ be a measure space and $(X, \bar{\Sigma}, \bar{\mu})$ be its completion measure space. Prove that
$\bar{\Sigma} = \{A \triangle B: A \in \Sigma, B \in \mathcal{N}\}$.
Recall that $\mathcal{N} = \{B \subset X: B \subset N, N \in \Sigma, \mu(N) = 0\}$ and $\bar{\Sigma} = \{A \cup B: A \in \Sigma, B \in \mathcal{N}\}$.

2. Completion ?-Algebras
Let (X, Σ, μ) be a measure space and (X, Σ̅, μ̅) be its completion measure space. Prove that
Σ̅ = {A B: A ∈Σ, B ∈𝒩}.
Recall that 𝒩 = {B ⊂ X: B ⊂ N, N ∈Σ, μ(N) = 0} and Σ̅ = {A ∪ B: A ∈Σ, B ∈𝒩}.

Added by Oscar V.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

04/10/2023

Step 1

Step 1: Recall that the completion measure space X' is defined as the set of all subsets A' of X such that A' = A∪B, where A∈E and B∈N. Show more…

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2. Completion of Algebras: Let X be a measure space and X' be its completion measure space. Prove that {AB: A∈E, B∈N} Recall that N = {B⊆X: B∈N, μ(B^c)=0} and A' = {A∪B: A∈E, B∈N}