# Monotone convergence theorem

In mathematics, there are several theorems dubbed monotone convergence; here we present some major examples.

1. If ak is a monotone sequence of real numbers (e.g., if ak ≤ ak+1,) then this sequence has a limit (if we admit plus and minus infinity as possible limits.) The limit is finite if and only if the sequence is bounded.
2. If for all natural numbers j and k, aj,k is a non-negative real number and aj,k ≤ aj+1,k, then
[itex]\lim_{j\to\infty} \sum_k a_{j,k} = \sum_k \lim_{j\to\infty} a_{j,k}[itex]
3. If fk are non-negative measurable real-valued functions with measure μ such that for each k and x, fk(x) ≤ fk+1(x), then
[itex]\lim_{k\to\infty} \int f_k(x)d\mu(x) = \int\lim_{k\to\infty} f_k(x)d\mu(x) [itex]
This theorem generalizes the previous one. It is sometimes called the Lebesgue monotone convergence theorem, and is probably the most important monotone convergence theorem.

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