Editing Monotone convergence theorem (section)

{{short description|Theorems on the convergence of bounded monotonic sequences}}
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In the mathematical field of [[real analysis]], the '''monotone convergence theorem''' is any of a number of related theorems proving the good [[convergence (mathematics)|convergence]] behaviour of [[monotonic sequence]]s, i.e. sequences that are non-[[increasing]], or non-[[decreasing]]. In its simplest form, it says that a non-decreasing  [[Bounded function|bounded]]-above  sequence of real numbers <math>a_1 \le a_2 \le a_3 \le ...\le K</math>  converges to its smallest upper bound, its [[supremum]]. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its [[infimum]]. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded.

For sums of non-negative increasing sequences <math>0 \le a_{i,1} \le a_{i,2} \le \cdots </math>, it says that taking the sum and the supremum can be interchanged.

In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in [[measure theory]] due to [[Lebesgue]] and [[Beppo Levi]] that says that for sequences of non-negative pointwise-increasing [[measurable function]]s <math>0 \le f_1(x) \le f_2(x) \le \cdots</math>, taking the integral and the supremum can be interchanged with the result being finite if either one is finite.