Ulam's Theorem

The set of Ulam is a mathematical theorem on the partial area of the measure theory , the on the mathematician Stanislaw Ulam back. The set deals with special properties of Borel dimensions on Polish spaces .

Formulation of the sentence

Ulam's theorem can be stated as follows:

Let be a Polish space and be a Borel measure on the σ-algebra of the Borel sets of . ${\displaystyle X}$ ${\displaystyle \mu \colon {\mathfrak {B}}(X)\to [0,\infty ]}$ ${\displaystyle X}$

Then:

(1) is a regular measure . ${\displaystyle \mu }$

(2) is a moderate level in the sense that ${\displaystyle \mu }$

that a representation as a countable union of form ${\displaystyle X}$

${\displaystyle X=\bigcup _{n\in \mathbb {N} }U_{n}}$

in which each is an open set of with . ${\displaystyle U_{n}\;(n\in \mathbb {N} )}$ ${\displaystyle X}$ ${\displaystyle \mu (U_{n})<\infty }$

Tightening

As Paul-André Meyer showed, Ulam's sentence can be tightened considerably by replacing the Polish rooms with the so-called Suslin rooms . A Suslin room is a Hausdorff room in such a way that there is also a Polish room with a constant surjection . ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle f\colon X\to S=f(X)}$

The set of Paul-André Meyer then says:

Every Borel measure on a Suslin room is regular and moderate . ${\displaystyle \mu \colon {\mathfrak {B}}(S)\to [0,\infty ]}$ ${\displaystyle S}$

The fact that this sentence intensifies Ulam's sentence arises from the fact that every Polish space under the identical mapping is always also a Suslin space. ${\displaystyle X}$

swell

• Jürgen Elstrodt : Measure and integration theory (=  Springer textbook - basic knowledge of mathematics ). 7th, corrected and updated edition. Springer-Verlag , Heidelberg (inter alia) 2011, ISBN 978-3-642-17904-4 .
• John C. Oxtoby , SM Ulam: On the existence of a measure invariant under a transformation . In: Ann. of Math. (2) . tape 40 , 1939, pp. 560-566 , JSTOR : 1968940 . MR0000097
• John C. Oxtoby: Invariant measures in groups which are not locally compact . In: Transactions of the American Mathematical Society . tape 60 , 1946, pp. 215-237 , JSTOR : 1990145 . MR0018188

Individual evidence

1. ↑ ^{a b c} Jürgen Elstrodt: Measure and integration theory. 2011, pp. 320-323
2. ↑ The union set resulting from a countable union is not necessarily itself countable.