Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 1

Question

Math 6010 • Linear Models Fall 2016 • Question 1

Construct an example of a countable family \(\{\mathcal {F}_i\}_{i=1}^{\infty }\) of \(\sigma \)-algebras such that \[ \sigma \left (\bigcup _{i=1}^{\infty }\mathcal {F}_i\right ) \ne \bigcup _{i=1}^{\infty }\mathcal {F}_i. \] Can you do this so that \(\mathcal {F}_i\subset \mathcal {F}_{i+1}\) for all \(i\geq 1\)? Justify your reasoning.

Raw LaTeX

Construct an example of a countable family
\(\{\mathcal{F}_i\}_{i=1}^{\infty}\) of \(\sigma\)-algebras such that
\[
  \sigma\left(\bigcup_{i=1}^{\infty}\mathcal{F}_i\right)
  \ne
  \bigcup_{i=1}^{\infty}\mathcal{F}_i.
\]
Can you do this so that \(\mathcal{F}_i\subset\mathcal{F}_{i+1}\) for all
\(i\geq1\)? Justify your reasoning.

Topic Tags

Uniform Integrability and Truncation Methods