Monotone convergence theorem

Called the set of the monotone convergence theorem of Beppo Levi also ( by Beppo Levi ), is an important sentence from the measure and integration theory, a branch of mathematics. He can make a statement as to let the conditions under which to exchange integration and thresholding.

Mathematical formulation

Let be a measure space. Is a sequence of non-negative, measurable functions μ - almost everywhere monotonically increasing converges to a measurable function, the following applies

Variant for falling consequences

If a function is a result of non-negative, measurable functions, the falling converges μ - almost everywhere to a measurable monotone function, also applies

Idea of ​​proof

The fact that the right side is less than or equal to the left, it follows from the monotonicity of the integral. For the proof significantly so is the other way: This can be about first show for simple functions and transferred from there to general measurable functions.

Probabilistic formulation

Let be a probability space and a non-negative almost surely increasing sequence of random variables, then for their expectation values

A similar statement is also true for conditional expectations: If a sub - algebra and integrable, then applies almost certainly

Application of the theorem on sets of functions

Again let be a measure space. Non-negative, measurable functions For each result

This follows by applying the rate to the sequence of partial sums. There are non-negative, is increasing.