Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2015 • Question 2

Question

Math 6210 • Real Analysis Fall 2015 • Question 2

(a) If \(f_i\to g\) are in \(L^1(\mathbb R,\lambda )\) and \(\phi \in C(\mathbb R)\), does \[ \lim _{i\to \infty }\int f_i\phi \,d\lambda =\int g\phi \,d\lambda ? \]

(b) Show that if \(f_i\to g\) in \(L^1(\lambda ,[0,1])\), then there exists a subsequence \(n_1,\ldots \) so that \(f_{n_i}\to g\) almost everywhere.

Raw LaTeX

(a) If \(f_i\to g\) are in \(L^1(\mathbb R,\lambda)\) and
\(\phi\in C(\mathbb R)\), does
\[
\lim_{i\to\infty}\int f_i\phi\,d\lambda=\int g\phi\,d\lambda?
\]

(b) Show that if \(f_i\to g\) in \(L^1(\lambda,[0,1])\), then there exists a
subsequence \(n_1,\ldots\) so that \(f_{n_i}\to g\) almost everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lp Spaces and Sequence Spaces