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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Fall 2024 • Question 1

Question

Math 6040 • Probability Fall 2024 • Question 1

Let \(q\) be an integrable random variable. Prove that for each \(\epsilon > 0\) we can pick \(\delta > 0\) so that \(\mu (A) < \delta \) implies \[ \int _{A}|q|\,d\mu < \epsilon . \] Let \(f \ge 0\). (Hint: Show first that for any \(f \ge 0\), \(\int f \wedge n\,d\mu \nearrow \int f\,d\mu \) as \(n \to \infty \).)

Raw LaTeX

Let \(q\) be an integrable random variable. Prove that for each \(\epsilon > 0\) we can pick \(\delta > 0\)
so that \(\mu(A) < \delta\) implies
\[
\int_{A}|q|\,d\mu < \epsilon.
\]
Let \(f \ge 0\).
(Hint: Show first that for any \(f \ge 0\), \(\int f \wedge n\,d\mu \nearrow \int f\,d\mu\) as \(n \to \infty\).)

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods