Vitali-Carathéodory theorem

The set of Vitali-Carathéodory is a mathematical theorem that in the transition area between the area of analysis and the field of measure theory is settled and the well-known analyst Walter Rudin both mathematicians Giuseppe Vitali and Constantin Carathéodory impute. Together with Lusin's theorem , it is one of the theorems about the continuity properties of measurable real-valued functions on certain measure spaces over locally compact Hausdorff spaces .

Formulation of the sentence

The sentence can be formulated as follows:

A locally compact Hausdorff space is given , provided with Borel's σ-algebra as well as a Borel measure that is regular from inside and outside ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}(X)}$

${\displaystyle \mu \colon {\mathcal {B}}(X)\rightarrow [0,\infty ]}$.

Next is given a - integrable real-valued function ${\displaystyle \mu }$

${\displaystyle f\colon X\rightarrow \mathbb {R} }$.

Then:

For every real number there is a pair of real-valued functions ${\displaystyle \epsilon >0}$

${\displaystyle u,v\colon X\rightarrow \mathbb {R} }$

with the following properties:

(1) is above semi-continuous and limited upwards . ${\displaystyle u}$

(2) is sub-semi-continuous and limited below . ${\displaystyle v}$

(3) . ${\displaystyle u(x)\leq f(x)\leq v(x)\;\;(\forall x\in X)}$

(4) . ${\displaystyle {\int _{X}}(v-u)\;\mathrm {d} \mu \;<\epsilon }$

Sources and background literature

• 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 .
• Walter Rudin : Real and Complex Analysis . 2nd, improved edition. Oldenbourg Wissenschaftsverlag , Berlin 2009, ISBN 978-3-486-59186-6 .

Individual evidence

1. ^ Walter Rudin: Real and Complex Analysis. 2009, p. 65 ff.
2. ↑ Rudin, op.cit., P. 66