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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1993 • Question 3

Question

Math 6210 • Real Analysis Fall 1993 • Question 3

Let \(X\) be as in Problem 2. Give an example of a sequence of measurable, nonnegative functions \(f_n:X\to \mathbb R\), \(f_1\ge f_2\ge \cdots \ge f_n\ge \cdots \), converging pointwise to zero, such that \(\int _X f_n(x)\,dx\) does not converge to zero.

Can any of the integrals \(\int _X f_n(x)\,dx\) be finite? Justify your answer.

Raw LaTeX

Let \(X\) be as in Problem 2. Give an example of a sequence of
measurable, nonnegative functions \(f_n:X\to\mathbb R\),
\(f_1\ge f_2\ge\cdots\ge f_n\ge\cdots\), converging pointwise to zero, such that
\(\int_X f_n(x)\,dx\) does not converge to zero.

Can any of the integrals \(\int_X f_n(x)\,dx\) be finite? Justify your answer.

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lp Spaces and Sequence Spaces Zeros of Holomorphic Functions and the Identity Theorem