Dominated convergence theorem

The set of the dominated convergence (also set by the majorisierenden convergence rate of the dominated convergence theorem of Lebesgue or ) is a central limit result in measure and integration theory, and goes back to the French mathematician Henri Léon Lebesgue.

The set provides a decision criterion for the commutativity of the integral and limit formation.

The formal statement of the theorem

Let be a measure space and is a consequence of - measurable functions.

The sequence converges - almost everywhere to a function - measurable. Furthermore, the result will outvoted by an integrable function, that is needed for everyone -almost everywhere. Note that in the local definition of the integration value is excluded, that is.

Then, and all - integrable and we have:

This also implies that

Comments on the conditions

• On the condition of Majorisierbarkeit can not be waived. Serves as an example by the sequence, defined where the indicator function call on. It is almost everywhere, yet is

• On the assumption that the function is measurable, can be dispensed with if instead it is known that a complete measure space is because the function is automatically measurable. As follows from the measurability, if it is known that the sequence anywhere, not only almost all converges.

Dominated convergence in spaces ( conclusion )

Let be a measure space, a real number and is a consequence of - measurable functions.

Continue the sequence converges almost everywhere to - a - measurable function, and the result will majorized by a function, ie, for all true -almost everywhere.

Then all and in and it is considered that the sequence converges to in the sense of, ie

Sketch of proof: Application of the original sentence on the sequence of functions with the majorant.

Dominated convergence for random variables

Since random variable nothing more than measurable functions on special measure spaces, namely, are the probability spaces can also be applied to a random variable of the theorem on the dominated convergence. Here you can even weaken the conditions on the result: It is sufficient that the sequence converges in probability instead of the stronger requirement of pointwise convergence almost everywhere:

Let be a probability space, a real number and let a sequence of real-valued random variables.

Continue the sequence converges in probability to a random variable and the result will outvoted by a random variable, ie applies to everyone -almost everywhere.

Then all and in and it is considered that the sequence converges to the meaning of and