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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2023 • Question 5

Question

Math 6210 • Real Analysis Fall 2023 • Question 5

Let \(f\) be Lebesgue integrable on \((0,1)\). For \(x\in (0,1)\) define \[ g(x)=\int _x^1 \frac {f(t)}{t}\,d\lambda . \]

Prove that \(g\) is Lebesgue integrable on \((0,1)\) and that \[ \int _0^1 g(x)\,d\lambda =\int _0^1 f(x)\,d\lambda . \]

Raw LaTeX

Let \(f\) be Lebesgue integrable on \((0,1)\). For \(x\in(0,1)\) define
\[
g(x)=\int_x^1 \frac{f(t)}{t}\,d\lambda.
\]

Prove that \(g\) is Lebesgue integrable on \((0,1)\) and that
\[
\int_0^1 g(x)\,d\lambda=\int_0^1 f(x)\,d\lambda.
\]

Topic Tags

Lebesgue Integration Lebesgue Measure and Translation Invariance