Theorem by Vitali-Hahn-Saks

The set of Vitali-Hahn-Saks is a set of the mathematical part area of the measure theory . It goes back to Giuseppe Vitali , Hans Hahn and Stanisław Saks and essentially states that the quantitative limit value of a series of signed measurements is such again.

First formulation of the sentence

Let it be a measurement space and then a sequence of signed measurements so that it converges for every measurable amount . Further, let it be a finite measure so that each is absolutely continuously against . Then the formula defines a signed measure that is also absolutely constantly against . ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\mu _{n}(E))_{n\in \mathbb {N} }}$ ${\displaystyle E\in \Sigma }$ ${\displaystyle \lambda }$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \mu _{n}}$ ${\displaystyle \lambda }$ ${\displaystyle \mu (E):=\lim _{n\to \infty }\mu _{n}(E)}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \lambda }$

The special content of this theorem is that the σ-additivity of is transferred to and that the absolute continuity against is retained. One can free oneself from the assumption about the existence of , because for each sequence is defined by a measure against which each is absolutely continuous. Here and are the variation or total variation norm of . Therefore one can formulate the above theorem without mentioning the absolute continuity and get the following theorem, also known as Nikodým's convergence theorem: ${\displaystyle \mu _{n}}$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle (\mu _{n}(E))_{n\in \mathbb {N} }}$ ${\displaystyle \textstyle \lambda (E):=\sum _{n=1}^{\infty }2^{-n}(1+\|\mu _{n}\|)^{-1}|\mu _{n}|(E)}$ ${\displaystyle \mu _{n}}$ ${\displaystyle |\mu _{n}|}$ ${\displaystyle \|\mu _{n}\|}$ ${\displaystyle \mu _{n}}$

Second formulation of the sentence

Let it be a measure space and on it a sequence of finite signed measures, so that for every measurable amount converges and is finite. Then the formula defines a signed dimension . ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\mu _{n}(E))_{n\in \mathbb {N} }}$ ${\displaystyle E\in \Sigma }$ ${\displaystyle \mu (E):=\lim _{n\to \infty }\mu _{n}(E)}$ ${\displaystyle (\Omega ,\Sigma )}$

This version is weaker because it no longer contains the maintenance of absolute continuity against a further dimension. Note that finiteness is a necessary condition, as the following example illustrates. Let and , where Borel's sigma algebra denotes on. For , define if , otherwise define . Then applies if there is no upper limit. Otherwise applies . is not a measure, since both but should also apply. ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Omega =\mathbb {R} }$ ${\displaystyle \Sigma ={\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle E\in {\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle \mu _{n}(E):=\infty }$ ${\displaystyle E\cap [n,\infty )\not =\emptyset }$ ${\displaystyle \mu (E):=0}$ ${\displaystyle \lim _{n\to \infty }\mu _{n}(E)=\infty }$ ${\displaystyle E}$ ${\displaystyle \lim _{n\to \infty }\mu _{n}(E)=0}$ ${\displaystyle \mu (E):=\lim _{n\to \infty }\mu _{n}(E)}$ ${\displaystyle \mu ([1,\infty ))=\infty }$ ${\displaystyle \mu ([1,\infty ))=\sum _{i=1}^{\infty }\mu ([i,i+1))=0}$

Applications

The space of the signed dimensions on a dimension space is a vector space that becomes a Banach space with total variation as the norm . An important application of the Vitali-Hahn-Saks theorem is to characterize the relatively weakly compact sets as precisely those bounded sets that are uniformly absolutely continuous to a finite measure. Another application is that weak is sequence- complete , that is, that every Cauchy sequence of the space that is uniform in weak topology converges weakly. ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle M(\Omega ,\Sigma )}$ ${\displaystyle M(\Omega ,\Sigma )}$ ${\displaystyle M(\Omega ,\Sigma )}$ ${\displaystyle M(\Omega ,\Sigma )}$

Individual evidence

1. ↑ G. Vitali: Sull'integrazione per serie , Rendiconti del Circolo Matematico di Palermo (1907), Volume 23, Pages 137-155
2. ↑ H. Hahn: About sequences of linear operations , monthly books for mathematics and physics (1922), volume 32, pages 3-88
3. ^ S. Saks: Addition to the Note on Some Functionals , Transactions of the American Mathematical Society (1933), Volume 35, pp. 965-970
4. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Appendix C, Sentence C.3
5. ↑ JL Doob: Measure Theory , Chapter IX, Section 10: Vitali-Hahn-Saks-theorem
6. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Appendix C, Sentence C.4
7. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Appendix C, Sentence C.7
8. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Appendix C, Sentence C.5