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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2022 • Question 3

Question

Math 6210 • Real Analysis Fall 2022 • Question 3

Let \(f : [0, 1] \to [0, \infty ]\) be measurable. Show that \(U(f) = \{(x, y) : 0 \le y \le f(x)\}\) is measurable and \[ \int f\,d\lambda = \lambda ^{2}(U(f)). \]

Raw LaTeX

Let \(f : [0, 1] \to [0, \infty]\) be measurable. Show that
\(U(f) = \{(x, y) : 0 \le y \le f(x)\}\) is measurable and
\[
\int f\,d\lambda = \lambda^{2}(U(f)).
\]

Topic Tags

Lebesgue Integration Measurable Functions