Let X be a set and let {Fi | i ∈ I} be an arbitrary collection of σ- algebras. Showthat the collection F := {F | ∀i ∈ I, F ∈ Fi} is a σ-algebra.

Question

Let X be a set and let {Fi | i ∈ I} be an arbitrary collection of σ- algebras. Showthat the collection F := {F | ∀i ∈ I, F ∈ Fi} is a σ-algebra.
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Solution 1

To show that F is a σ-algebra, we need to verify three properties:

  1. The empty set and X are in F.
  2. If A is in F, then its complement is in F.
  3. If {An} is a countable collection of sets in F, then the union of {An} is in F.

Let's verify each property:

  1. Since each Fi is a σ-algebra, the emp Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob
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Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
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