Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2024 • Question 3

Question

Math 6210 • Real Analysis Fall 2024 • Question 3

Prove the following generalization of the Dominated Convergence Theorem. Let \((X,\mathcal {M},\mu )\) be a measure space. Suppose \(f_n,g_n,f,g:X\to [0,\infty )\) are measurable and have finite integrals, \(f_n\to f\), \(g_n\to g\) (pointwise convergence), \(f_n\le g_n\), \(\int g_n\to \int g<\infty \). Then \(\int f_n\to \int f\).

Raw LaTeX

Prove the following generalization of the Dominated Convergence Theorem. Let \((X,\mathcal{M},\mu)\) be a measure space. Suppose \(f_n,g_n,f,g:X\to[0,\infty)\)
are measurable and have finite integrals, \(f_n\to f\), \(g_n\to g\) (pointwise convergence),
\(f_n\le g_n\), \(\int g_n\to\int g<\infty\). Then \(\int f_n\to\int f\).

Topic Tags

Absolute Continuity and Singular Measures Almost Everywhere and Convergence in Measure Convergence Theorems Measure Theory