Monotone convergence theorem

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

If a_k is a monotone sequence of real numbers (e.g., if a_k ≤ a_k+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.
If for all natural numbers j and k, a_j,k is a non-negative real number and a_j,k ≤ a_j+1,k, then
$\lim_{j\to\infty} \sum_k a_{j,k} = \sum_k \lim_{j\to\infty} a_{j,k}$
If f_k are non-negative measurable real-valued functions with measure μ such that for each k and x, f_k(x) ≤ f_k+1(x), then
$\lim_{k\to\infty} \int f_k(x)d\mu(x) = \int\lim_{k\to\infty} f_k(x)d\mu(x)$
This theorem generalizes the previous one. It is sometimes called the Lebesgue monotone convergence theorem, and is probably the most important monotone convergence theorem.