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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2025 • Question 2

Question

Math 6210 • Real Analysis Spring 2025 • Question 2

Let \(\mu \) be a Borel measure on \(\mathbb {R}\) (i.e. a measure defined on the Borel \(\sigma \)-algebra). Suppose it satisfies:

  1. \[ \mu ([0,1])=1. \]
  2. \(\mu (A+t)=\mu (A)\) for every Borel set \(A\) and every \(t\in \mathbb {R}\), i.e. \(\mu \) is translation invariant.

Show that \(\mu \) equals the Lebesgue measure.

Raw LaTeX

Let \(\mu\) be a Borel measure on \(\mathbb{R}\) (i.e. a measure defined on the Borel
\(\sigma\)-algebra). Suppose it satisfies:
\begin{enumerate}

\item[(i)]

\[
\mu([0,1])=1.
\]

\item[(ii)]
\(\mu(A+t)=\mu(A)\) for every Borel set \(A\) and every \(t\in\mathbb{R}\), i.e. \(\mu\) is
translation invariant.

\end{enumerate}

Show that \(\mu\) equals the Lebesgue measure.

Topic Tags

Absolute Continuity and Singular Measures Borel Sets and Sigma-Algebras Lebesgue Measure and Translation Invariance Measure Theory