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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2021 • Question 4

Question

Math 6210 • Real Analysis Fall 2021 • Question 4

Let \(f : [0, 1] \to \mathbb {R}\) be a continuous function and define a signed measure \(\mu \) on the Lebesgue measurable subsets of \([0, 1]\) by

\[ \mu (E) = \int _{E} f\,dm \]

where \(m\) is Lebesgue measure. Suppose that for all \(n = 0, 1, 2, 3, \ldots \)

\[ \int _{[0,1]} x^{n}\,d\mu = 0 \]

Prove that \(\mu = 0\).

Raw LaTeX

Let \(f : [0, 1] \to \mathbb{R}\) be a continuous function and define a signed measure
\(\mu\) on the Lebesgue measurable subsets of \([0, 1]\) by

\[
\mu(E) = \int_{E} f\,dm
\]

where \(m\) is Lebesgue measure. Suppose that for all \(n = 0, 1, 2, 3, \ldots\)

\[
\int_{[0,1]} x^{n}\,d\mu = 0
\]

Prove that \(\mu = 0\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Lebesgue Measure and Translation Invariance Measure Theory