Monotone convergence theorem

There is a variety of theorems dubbed monotone convergence, here we present a few main examples.

1) 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 bounded if and only if the sequence is bounded.

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

lim_j∑_ka_j,k=∑_k lim_ja_j,k

3) 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∫f_k(x)dμ(x)=∫lim_kf_k(x)dμ(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.