Monotone convergence theorem

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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 behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers ${\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq ...\leq K}$ 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 ${\displaystyle 0\leq a_{i,1}\leq a_{i,2}\leq \cdots }$, 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 functions ${\displaystyle 0\leq f_{1}(x)\leq f_{2}(x)\leq \cdots }$, taking the integral and the supremum can be interchanged with the result being finite if either one is finite.