“monotone convergence theorem”的概念、定义、翻译、参考文献-科学参考

monotone convergence theorem

Mathematics

Let f_n:ℝ → ℝ, n ≥ 1, be a sequence of Lebesgue integrable functions (see Lebesgue integration) which is increasing—that is, f_n + 1 (x) ≥ f_n(x) for all n,x—and such that the integrals $\int_{- \infty}^{\infty} f_{n} (x) d x$ are bounded above. Then there exists an integrable function f such that f_n(x) converges to f(x) almost everywhere and

$\int_{- \infty}^{\infty} f (x) d x = lim_{n \to \infty} \int_{- \infty}^{\infty} f_{n} (x) d x .$

Compare dominated convergence theorem.

单词	monotone convergence theorem
释义	monotone convergence theorem Mathematics Let f_n:ℝ → ℝ, n ≥ 1, be a sequence of Lebesgue integrable functions (see Lebesgue integration) which is increasing—that is, f_n + 1 (x) ≥ f_n(x) for all n,x—and such that the integrals $\int_{- \infty}^{\infty} f_{n} (x) d x$ are bounded above. Then there exists an integrable function f such that f_n(x) converges to f(x) almost everywhere and $\int_{- \infty}^{\infty} f (x) d x = lim_{n \to \infty} \int_{- \infty}^{\infty} f_{n} (x) d x .$ Compare dominated convergence theorem.
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