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
- limj∑kaj,k=∑k limjaj,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
- limk∫fk(x)dμ(x)=∫limkfk(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.
See also infinite series