• DCT states limit and integral commute under pointwise a.e. convergence with integrable dominating function. • Requires measurable functions and existence of integrable g bounding |f_n| for all n. • Counterexample: f_n(x)=nχ_(0,1/n] shows convergence to 0 but no integrable dominator. • Without domination, interchange of limit and integral can fail, as shown by counterexample. • DCT guarantees integrability of limit function f and equality of integral limits. • Useful for evaluating limits of integrals, e.g., ∫ n sin(x/n)/(x(x^2+1)) dx. • Related theorems: Monotone Convergence Theorem and Fatou’s Lemma provide alternative conditions. • DCT is central in Lebesgue integration theory for handling pointwise convergence.
Article Summaries:
- The article explains the Dominated Convergence Theorem (DCT), a key result in Lebesgue integration that allows interchange of limit and integral when a sequence of measurable functions is dominated by an integrable function. It contrasts this with the Monotone Convergence Theorem and Fatou’s Lemma, noting that without domination the interchange can fail. A counterexample using (f_n(x)=n\chi_{(0,1/n]}) shows the failure: each (f_n) integrates to 1 while the pointwise limit is 0. The article then applies the DCT to compute (\lim_{n\to\infty}\int_{\mathbb R}\frac{n\sin(x/n)}{x(x^2+1)}dx), demonstrating that the limit equals (\int_{\mathbb R}\frac{1}{1+x^2}dx).
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