Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2016 • Question 5

Question

Math 6210 • Real Analysis Summer 2016 • Question 5

Prove that if \(\mu \) is a non-atomic Borel probability measure on \(\mathbb R\), then the function \(f:\mathbb R\to \mathbb R\) defined by \[ f(t)=\mu ((-\infty ,t)) \] is continuous.

Raw LaTeX

Prove that if \(\mu\) is a non-atomic Borel probability measure on
\(\mathbb R\), then the function \(f:\mathbb R\to\mathbb R\) defined by
\[
f(t)=\mu((-\infty,t))
\]
is continuous.

Topic Tags

Absolute Continuity and Singular Measures Measure Theory Outer Measures and Caratheodory Measurability Product Measures and Fubini-Tonelli Theorems