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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2014 • Question 3

Question

Math 6210 • Real Analysis Fall 2014 • Question 3

Let \(f_i:[0,1]\to \mathbb R\) be a sequence of functions that converge pointwise to \(f_\infty \).

(a) Show that if the \(f_i\) are uniformly bounded then \[ \lim _{i\to \infty }\int f_i\,d\lambda =\int f_\infty \,d\lambda . \]

(b) What if \(\lVert f_i\rVert _1\le 1\) for all \(i\)?

(c) What if \(\lVert f_i\rVert _2\le 1\) for all \(i\)?

Raw LaTeX

Let \(f_i:[0,1]\to\mathbb R\) be a sequence of functions that
converge pointwise to \(f_\infty\).

(a) Show that if the \(f_i\) are uniformly bounded then
\[
\lim_{i\to\infty}\int f_i\,d\lambda=\int f_\infty\,d\lambda.
\]

(b) What if \(\lVert f_i\rVert_1\le1\) for all \(i\)?

(c) What if \(\lVert f_i\rVert_2\le1\) for all \(i\)?

Topic Tags

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