Vitali's convergence theorem

The convergence rate of Vitali , also Vitali criterion or Vitalis criterion for convergence, ${\displaystyle L^{p}}$ is a set of measure theory , the for function sequences indicating criteria under which the convergence in the p-th agent and the convergence locally to measure are equivalent. Criteria for the convergence according to measure and its stochastic equivalent, the convergence in probability , can also be derived from this. The proposition is named after Giuseppe Vitali , who proved it in 1907.

statement

Given is a measure space and , where or is. Be , also be . Then are equivalent: ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle f,f_{n}\colon X\to \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle p\in (0,\infty )}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} },f\in {\mathcal {L}}^{p}(X,{\mathcal {A}},\mu )}$

1. They converge in the p-th mean to ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$
2. The converge locally to measure against and are uniformly integrable p-th agent . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$

comment

The statement also applies if the convergence locally by measure is replaced by the convergence according to measure , because every sequence convergent in the p-th mean is due

${\displaystyle \mu (\{|f_{n}-f|\geq \varepsilon \})\leq {\tfrac {1}{\varepsilon ^{p}}}\int _{X}|f_{n}-f|^{p}\mathrm {d} \mu ={\tfrac {1}{\varepsilon ^{p}}}\Vert f_{n}-f\Vert _{p}^{p}}$

custom-made convergent. In addition, according to the above theorem, it is also locally convergent according to measure and integrable to the same degree in the p-th mean, so it is only integrable to the same degree in the p-th mean. Thus, from the convergence in the p-th mean follows the uniform integrability and the convergence according to measure.

The convergence follows from the fact that the convergence to measure leads to local convergence to measure. Thus, a sequence that is convergent to measure and can be uniformly integrated in the p-th mean is also locally convergent to measure and integrable to the same degree in the p-th mean and thus also convergent in the pth mean according to the above sentence.

Examples

The following two examples show that if either the convergence locally according to measure or the equal integrability is dispensed with, the conclusion on the convergence in the p-th mean is incorrect.

Convergent locally made to measure, but not equally integrable

The sequence of functions is first defined with and the dimension space ${\displaystyle p=1}$ ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda |_{[0,1]})}$

${\displaystyle f_{n}=n^{2}\chi _{[0,1/n]}}$.

This converges locally by measure to 0, because for is ${\displaystyle \varepsilon \in (0,1]}$

${\displaystyle \lim _{n\to \infty }\lambda (\{n^{2}\chi _{[0,1/n]}\geq \varepsilon \})=\lim _{n\to \infty }{\frac {1}{n}}=0}$.

But it cannot be integrated equally (in the first mean) because it is

${\displaystyle \inf _{a\in [0,\infty )}\sup _{n\in \mathbb {N} }\int _{\{a<|f_{n}|\}}|f_{n}|\mathrm {d} \lambda =\infty }$.

Consequently the sequence of functions is also not (in the first mean) convergent to 0, because it is

${\displaystyle \lim _{n\to \infty }\int _{[0,1]}|f_{n}|\mathrm {d} \lambda =\lim _{n\to \infty }n^{2}\cdot {\frac {1}{n}}=\infty }$.

Equally integrable, but not convergent locally made to measure

Again, as above, you set and choose as the measurement space . The sequence of functions is defined by ${\displaystyle p=1}$ ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda |_{[0,1]})}$

${\displaystyle f_{n}:={\begin{cases}\chi _{[0;1/2]}&{\text{ für }}n{\text{ gerade }}\\\chi _{(1/2;1]}&{\text{ für }}n{\text{ ungerade }}\end{cases}}}$.

This sequence of functions can be integrated to the same degree in the first mean, since it is dominated by the integrable function, which is constant 1. Due to its oscillating behavior, however, the sequence cannot converge locally to measure, because there is no function for the basic set and , so that it becomes small. With an analogous argument it then also follows that the function sequence does not converge in the first mean. ${\displaystyle \varepsilon <{\tfrac {1}{2}}}$ ${\displaystyle f}$ ${\displaystyle \lambda (\{f_{n}-f\leq \varepsilon \})}$

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .