Monotone class theorem

The theorem on monotone classes is a central set of measure theory, the branch of mathematics that deals with the properties of measure spaces and functions on them.

Definition of a monotone vector space

Before the sentence can be formulated, we must first introduce the notion of monotone vector space. A lot of bounded, real-valued functions on an arbitrary space is called monotone if the following properties are satisfied:

• Is a vector space over the real numbers.
• All constant functions lie in.
• For every sequence of functions that satisfies and ( pointwise convergence ) with limited, the following applies:.

The theorem on monotone classes

It is a multiplicative ( ie under multiplication completed ) class of bounded, real-valued functions on a set and the class generated by the σ - algebra. Is also a monotonous vector space which contains, as a subset. Then stating the theorem on monotone classes that also contains all the bounded, measurable functions.

Applications

A classic application of the theorem on monotone classes is the proof of the theorem of Fubini. In some cases, you can also use the more descriptive evidence standard methods of integration of simple functions and applying the rate of the monotone convergence proved.