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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1999 • Question 3

Question

Math 6210 • Real Analysis 1999 • Question 3

Let \(D\) be the closed unit disc in \(\mathbb C\) and \(T\) the unit circle. Prove that for every \(\mu \) positive measure of total mass one on \(D\) there is a positive measure \(\nu \) of total mass one on \(T\) such that \[ \int f\,d\mu =\int f\,d\nu \] for every \(f\) which is continuous on \(D\) and holomorphic on the interior of \(D\).

Raw LaTeX

Let \(D\) be the closed unit disc in \(\mathbb C\) and \(T\) the unit
circle. Prove that for every \(\mu\) positive measure of total mass one on \(D\)
there is a positive measure \(\nu\) of total mass one on \(T\) such that
\[
\int f\,d\mu=\int f\,d\nu
\]
for every \(f\) which is continuous on \(D\) and holomorphic on the interior of \(D\).

Topic Tags

Absolute Continuity and Singular Measures Holomorphic Functions and Cauchy-Riemann Equations Lebesgue Measure and Translation Invariance Measure Theory