Theoretical induction

In the measure-theoretic induction (also algebraic induction called) is a method of proof from the measure theory , which is used for mathematical statements for a given set of measurable functions to show.

The basic idea behind the procedure is not to show the statement initially for all functions from the set, but to restrict oneself to a subset for which the statement is easy to prove. Then, successively larger and larger subsets are considered and the statement is also proven for these. For each step, use is made of the fact that the statement for the quantities from the previous steps has already been shown. After three or four steps, the statement is finally proven for all functions.

The method also plays an important role in probability theory and other areas of application of measure theory.

The proof method

A set of measurable functions is given . The claim is that the mathematical statement is true for everyone . The method usually consists of four steps. Sometimes step 1 is skipped so that induction is carried out with a total of three steps. ${\displaystyle {\mathcal {G}}}$ ${\displaystyle A(f)}$ ${\displaystyle f\in {\mathcal {G}}}$

1. Step: The statement is true for any measurable characteristic function from . ${\displaystyle A(f)}$ ${\displaystyle {\mathcal {G}}}$
2. Step: The statement is true for any positive simple function from . ${\displaystyle A(f)}$ ${\displaystyle {\mathcal {G}}}$
3. Step: The statement applies to any positive measurable function . ${\displaystyle A(f)}$ ${\displaystyle {\mathcal {G}}}$
4. Step: The statement applies to any measurable function . ${\displaystyle A(f)}$ ${\displaystyle {\mathcal {G}}}$

With each step, the amount of functions for which the statement is already considered, successively larger until the fourth and last step, the message is eventually detected for all functions.

Note that the proof of a step also implies all of the previous steps. This obviously applies, since every function considered in one step is also considered in all subsequent steps. For example, every positive simple function in step 2 is especially positive and measurable and therefore also part of step 3. For this reason, if the method is used sensibly, it is usually necessary to use the statement for the quantities when proving a step from the previous steps has already been shown.

Examples

example 1

The set of Fubini can be proved by means of maßtheoretischer induction.

Example 2

We consider the random variable and the amount . This means the smallest algebra that can be measured. We now consider the following statement: ${\displaystyle Y:\Omega \to \mathbb {R} }$ ${\displaystyle {\mathcal {G}}=\{X:\Omega \to \mathbb {R} \,|\,X\,{\text{ist }}\sigma (Y)\,{\text{messbar}}\}}$ ${\displaystyle \sigma (Y)}$ ${\displaystyle \sigma }$ ${\displaystyle Y}$

${\displaystyle A\,:\,{\text{Für alle}}\,X\in {\mathcal {G}}\,{\text{existiert eine messbare Funktion}}\,\phi :\mathbb {R} \to \mathbb {R} \,{\text{mit}}\,X=\phi (Y)}$

We shall now show the proposition with the aid of induction by mass theory. This applies to all steps . ${\displaystyle X\in {\mathcal {G}}}$

1st step: Be a characteristic function. Then applies with . After defining measurability and choosing the set , the existence of a set follows with ( Borel's σ-algebra ) and the property . Define now . Then follows for any : . ${\displaystyle X}$ ${\displaystyle X=\chi _{B}}$ ${\displaystyle B\in \sigma (Y)}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle C\in {\mathcal {B}}}$ ${\displaystyle B=Y^{-1}(C)}$ ${\displaystyle \phi =\chi _{C}}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle X(\omega )=\chi _{B}(\omega )=\chi _{C}(Y(\omega ))=\phi (Y(\omega ))}$

Step 2: Now be a positive simple function. Then with and . With the choice of and the assertion follows: ${\displaystyle X}$ ${\displaystyle X=\sum _{n=1}{c_{n}\cdot \chi _{A_{n}}}}$ ${\displaystyle c_{n}\in \mathbb {R^{+}} }$ ${\displaystyle A_{n}\in \sigma (Y)}$ ${\displaystyle \phi =\sum _{n=1}{c_{n}\cdot \chi _{B_{n}}}}$ ${\displaystyle A_{n}=Y^{-1}(B_{n})}$

${\displaystyle X(\omega )=(\sum _{n=1}{c_{n}\cdot \chi _{A_{n}})}(\omega )=\sum _{n=1}{c_{n}\cdot \chi _{A_{n}}(\omega )}=\sum _{n=1}{c_{n}\cdot \chi _{B_{n}}(Y(\omega ))}=(\sum _{n=1}{c_{n}\cdot \chi _{B_{n}}(Y))}(\omega )=\phi (Y(\omega ))}$

3rd step: Now consider an arbitrary and a sequence of positive simple functions, monotonically increasing against converging, ie , and for all and for almost all . Then for each using step 2, that is an appropriate one . Now set if the limit value exists and otherwise. Then follows: ${\displaystyle X\geq 0}$ ${\displaystyle (X_{n})_{n}}$ ${\displaystyle X}$ ${\displaystyle \lim _{n\to \infty }X_{n}=X}$ ${\displaystyle X_{n}\leq X_{n+1}}$ ${\displaystyle X_{n}\leq X}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle X_{n}}$ ${\displaystyle X_{n}=\phi _{n}(Y)}$ ${\displaystyle \phi _{n}(Y)}$ ${\displaystyle \phi (x)=\lim _{n\to \infty }\phi _{n}(x)}$ ${\displaystyle \phi (x)=0}$

${\displaystyle X(\omega )=\lim _{n\to \infty }X_{n}(\omega )=\lim _{n\to \infty }\phi _{n}(Y(\omega ))=\phi (Y(\omega ))}$

4th step: Now be arbitrary. Then there is and with . According to step 3 there is also a with and a with . Now set and the claim follows. ${\displaystyle X}$ ${\displaystyle X^{+}\geq 0}$ ${\displaystyle X^{-}\geq 0}$ ${\displaystyle X=X^{+}-X^{-}}$ ${\displaystyle \phi _{+}}$ ${\displaystyle X^{+}=\phi _{+}(Y)}$ ${\displaystyle \phi _{-}}$ ${\displaystyle X^{-}=\phi _{-}(Y)}$ ${\displaystyle \phi =\phi _{+}-\phi _{-}}$

literature

• Klaus D. Schmidt: Measure and probability , Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-21025-9 , page 109
• Hartmut Milbrodt: Probability Theory: An Introduction with Applications and Examples from Insurance and Financial Mathematics , VVW GmbH, Karlsruhe 2010, ISBN 978-3-89952-318-8 , page 286,287