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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Spring 2022 • Question 4

Question

Math 6040 • Probability Spring 2022 • Question 4

Let \(u\) be an integrable Borel function on \([0, 1)\) relative to the Lebesgue measure. For each \(n \ge 1\) and \(x \in [0, 1)\), let \(I_{n}(x) = [k2^{-n}, (k + 1)2^{-n})\) be the interval that contains \(x\) as \(k\) varies from 0 to \(2^{n} - 1\). Show that for Lebesgue-a.e. \(x\), \[ \lim _{n \to \infty } 2^{n}\int _{I_{n}(x)} u(y)\,dy = u(x). \]

(Hint: Cast this in a suitable martingale framework then use the result of problem 3.)

Raw LaTeX

Let \(u\) be an integrable Borel function on \([0, 1)\) relative to the Lebesgue measure. For
each \(n \ge 1\) and \(x \in [0, 1)\), let \(I_{n}(x) = [k2^{-n}, (k + 1)2^{-n})\) be the interval that contains
\(x\) as \(k\) varies from 0 to \(2^{n} - 1\). Show that for Lebesgue-a.e. \(x\),
\[
\lim_{n \to \infty} 2^{n}\int_{I_{n}(x)} u(y)\,dy = u(x).
\]


(Hint: Cast this in a suitable martingale framework then use the result of problem 3.)

Topic Tags

Exponential Martingales Lebesgue Differentiation via Martingales Martingales and Supermartingales Uniform Integrability and Truncation Methods