proof of dominated convergence theorem
Define the functions and as follows:
These suprema and infima exist because, for every , . These functions enjoy the following properties:
For every ,
For every ,
Each is measurable.
The first property follows from immediately from the definition of supremum. The second property follows from the fact that the supremum or infimum is being taken over a larger set to define than to define when . The third property is a simple consequence of the fact that, for any sequence of real numbers, if the sequence converges, then the sequence has an upper limit and a lower limit which equal each other and equal the limit. As for the fourth statement, it means that, for every real number and every integer , the sets
are measurable. However, by the definition of , these sets can be expressed as
respectively. Since each is assumed to be measurable, each set in either union is measurable. Since the union of a countable number of measurable sets is itself measurable, these unions are measurable, and hence the functions are measurable.
Because of properties 1 and 4 above and the assumption that is integrable, it follows that each is integrable. This conclusion and property 2 mean that the monotone convergence theorem is applicable so one can conclude that is integrable and that
By property 3, the right hand side equals .
By construction, and hence
Because the outer two terms in the above inequality tend towards the same limit as , the middle term is squeezed into converging to the same limit. Hence
|Title||proof of dominated convergence theorem|
|Date of creation||2013-03-22 14:33:58|
|Last modified on||2013-03-22 14:33:58|
|Last modified by||rspuzio (6075)|