Dominated convergence theorem

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In mathematics, Lebesgue's dominated convergence theorem states that if a sequence { f_n : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is "dominated" (explained below) by some nonnegative function g in <math>L^1</math>, then

<math>\int_S\lim_{n\rightarrow\infty} f_n=\lim_{n\rightarrow\infty}\int_S f_n.</math>

To say that the sequence is "dominated" by g means that

<math>|f_n(x)| \leq g(x)</math>

for every n and almost every x (i.e., the measure of the set of exceptional values of x is zero). By g in <math>L^1</math>, we mean

<math>\int_S\left|g\right|<\infty.</math>

So the theorem provides a sufficient condition under which integration and passing to the pointwise limit commute.

That the assumption that the sequence is dominated by some g in <math>L^1</math> can not be dispensed with may be seen as follows: let f_n(x) = n if 0 < x < 1/n and f_n(x) = 0 otherwise. Any g which dominates the sequence must also dominate h given by h(x) = sup_n f(x) for x > 0 (and 0 otherwise). Since

<math>\int_0^1 h(x)\,dx = \infty,</math>.

The order property of the Lebesgue integral tells us that there exists no function in <math>L^1</math> which dominate the sequence. A direct calculation shows integration and point-wise limit do not commute for this sequence:

<math>\int_0^1\lim_{n\rightarrow\infty} f_n(x)\,dx=0\neq 1=\lim_{n\rightarrow\infty}\int_0^1 f_n(x)\,dx.</math>

In contrast, Lebesgue's monotone convergence theorem does not require a sequence being dominated by a integrable g and instead assumes the given sequence is monotone. It hence states:

<math>\lim_{k\to\infty} \int f_k(x)\,d\mu(x) = \int\lim_{k\to\infty} f_k(x)\,d\mu(x) </math>

whenever <math> \{ f_n \}</math> is a monotone sequence of (real-valued) measurable functions.

Either theorem, dominated convergence theorem or monotone convergence theorem, can be shown as a corollary of the Fatou-Lebesgue theorem.he:משפט ההתכנסות הנשלטת ru:Теорема Лебега (мажорированная сходимость)