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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2010 • Question 3

Question

Math 6420 • PDE Spring 2010 • Question 3

Let \[ f(x)= \begin {cases} \frac {100}{n}, & x\ \text {rational},\ x=\frac {m}{n},\\ -2, & x\ \text {irrational}. \end {cases} \] Does the Riemann integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Does the Lebesgue integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Why is the Lebesgue integral used in defining the Hilbert space \(L^2[0,1]\), and not the Riemann integral?

Raw LaTeX

Let
\[
f(x)=
\begin{cases}
\frac{100}{n}, & x\ \text{rational},\ x=\frac{m}{n},\\
-2, & x\ \text{irrational}.
\end{cases}
\]
Does the Riemann integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Does the Lebesgue integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Why is the Lebesgue integral used in defining the Hilbert space \(L^2[0,1]\), and not the
Riemann integral?

Topic Tags

Cauchy Integral Formula and Integral Theorems Harmonic Functions and Poisson Integral Formula Hilbert Spaces and Orthogonality Linear Programming and Optimization