Theorem about monotonous classes

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

Definition of a monotonic vector space

Before the theorem can be formulated, we must first introduce the concept of a monotonic vector space. A set of bounded , real-valued functions on any space is called monotonic if the following properties are met: ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \;\Omega }$

• ${\displaystyle {\mathcal {H}}}$is a vector space over the real numbers .
• All constant functions are in . ${\displaystyle {\mathcal {H}}}$
• For every sequence of functions in that and ( pointwise convergence ) with met limited, then: . ${\displaystyle (f_{n})_{n\geq 1}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle 0\leq f_{1}\leq \cdots \leq f_{n}\leq \cdots }$ ${\displaystyle f_{n}\to f\;}$ ${\displaystyle f}$ ${\displaystyle f\in {\mathcal {H}}}$

The theorem about monotonous classes

Let it be a multiplicative (i.e. closed under multiplication) class of bounded, real-valued functions on a set and the σ-algebra generated by the class . In addition, let it be a monotonic vector space that contains as a subset . Then the theorem about monotonous classes says that it also contains all bounded, measurable functions. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}=\sigma ({\mathcal {M}})}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {A}}}$

Applications

A classic application of the theorem about monotonic classes is the proof of Fubini's theorem . In some cases proofs can also be proved with the more descriptive standard procedure of integrating simple functions and applying the theorem of monotonic convergence .

literature

• Claude Dellacherie, Paul-André Meyer : Probabilities and Potential (= North Holland Mathematics Studies. Vol. 29). North-Holland et al., Amsterdam et al. 1978, ISBN 0-7204-0701-X .
• Philip E. Protter: Stochastic integration and differential equations. Version 2.1 (= Applications of Mathematics. Stochastic Modeling and Applied Probability. Vol. 21). 2nd edition, corrected. 3rd printing. Springer, Berlin et al. 2005, ISBN 3-540-00313-4 .