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Monotone convergence theorem

There is a variety of theorems dubbed monotone convergence, here we present a few main 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 bounded if and only if the sequence is bounded.

2) If for each natural numbers j and k, aj,k is a non-negative real number, and furthermore, for each j,k, aj,k≤aj+1,k, then

limjkaj,k=∑k limjaj,k

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
This theorem generalizes the previous one. It is sometimes called the "Lebesgue monotone convergence theorem" and is probably the most important monotone convergence theorem.

See also infinite series