Equal integrability

The uniform integrability , also called uniform integrability , is a reinforcement of the concept of integrability in mathematics . In contrast to integrability, it is always a property of a family of functions and not that of a single function. However, the family can also be one element. The equilibrium integrability is particularly important in probability theory and measure theory , as it enables, using Vitali's convergence theorem, to establish a connection from the convergence in the p-th mean to the convergence in probability of the probability theory and the convergence according to measure or the Beat convergence locally according to the measure of measure theory. A family of functions can clearly be integrated evenly if the integral over “small” quantities does not assume too large values.

definition

Let be a measure space with measure and be the set of functions that can be integrated with respect to this measure. The positive part of a function is denoted by. ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {L}}^{1}(\mu )}$ ${\displaystyle f^{+}:=\max\{0,f\}}$

For general measurements

A family of functions is said to be equally integrable if it fulfills one of the following equivalent definitions: ${\displaystyle {\mathcal {F}}\subset {\mathcal {L}}^{1}(\mu )}$

• The following applies to all positive functions ${\displaystyle g\in {\mathcal {L}}^{1}(\mu )}$

${\displaystyle \inf _{g\geq 0}\sup _{f\in {\mathcal {F}}}\int (\vert f\vert -g)^{+}\mathrm {d} \mu =\inf _{g\geq 0}\sup _{f\in {\mathcal {F}}}\int _{\{|f|>g\}}\vert f\vert \mathrm {d} \mu =0}$.

• The following two conditions are met:

1. It is ${\displaystyle \sup _{f\in {\mathcal {F}}}\int \vert f\vert \mathrm {d} \mu <\infty }$
2. There is an integrable function , so that for any one there is one, so that for every quantity with ${\displaystyle 0\leq h\in {\mathcal {L}}^{1}(\mu )}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta (\varepsilon )>0}$ ${\displaystyle A\in {\mathcal {A}}}$

${\displaystyle \int _{A}h\;\mathrm {d} \mu <\delta (\varepsilon )}$

holds that

${\displaystyle \sup _{f\in {\mathcal {F}}}\int _{A}|f|\;\mathrm {d} \mu \leq \varepsilon }$

is.

For finite dimensions

If the measure is finite, so is , then the definitions simplify. The family is said to be equally integrable if one of the following equivalent definitions applies: ${\displaystyle \mu (\Omega )<\infty }$

• It is

${\displaystyle \inf _{a\in [0,\infty )}\sup _{f\in {\mathcal {F}}}\int (|f|-a)^{+}\mathrm {d} \mu =0}$

• It is

${\displaystyle \inf _{a\in [0,\infty )}\sup _{f\in {\mathcal {F}}}\int _{\{a<|f|\}}|f|\mathrm {d} \mu =0}$

• The following two conditions are met:

1. It is . ${\displaystyle \sup _{f\in {\mathcal {F}}}\int \vert f\vert \mathrm {d} \mu <\infty }$
2. For anything one exists , so that for everyone with ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta (\varepsilon )>0}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \mu (A)<\delta (\varepsilon )}$

${\displaystyle \sup _{f\in {\mathcal {F}}}\int _{A}|f|\mathrm {d} \mu \leq \varepsilon }$

applies.

Equally integrable in the p-th mean

A family of functions is said to be equally integrable in the p-th mean , if the family is equally integrable. ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle (|f_{i}|^{p})_{i\in I}}$

properties

• Every finite set can be integrated equally. ${\displaystyle {\mathcal {F}}\subset {\mathcal {L}}^{1}(\mu )}$
• Be families of functions and be equally integrable. Exists for each one , so that , is also integrable equally. ${\displaystyle {\mathcal {F}},{\mathcal {G}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle g\in {\mathcal {G}}}$ ${\displaystyle f\in {\mathcal {F}}}$ ${\displaystyle \vert g\vert \leq \vert f\vert }$ ${\displaystyle {\mathcal {G}}}$
• If one exists , so that for all , it is equally integrable. This is a direct special case of the two above properties ${\displaystyle g\in L^{1}(\mu )}$ ${\displaystyle |f|\leq |g|}$ ${\displaystyle f\in {\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}}$
• A sequence of measurable functions converges on the mean , i.e. with respect to the norm against a function , when it converges in terms of measure and is equally integrable. This follows from Vitali's convergence theorem . ${\displaystyle f_{n}}$ ${\displaystyle L^{1}}$ ${\displaystyle f}$
• General converge a sequence of measurable functions if and in pth mean , that with respect to the norm to a function when the dimensions to be uniformly integrable converges in pth mean. This statement also follows from Vitali's convergence theorem. ${\displaystyle f_{n}}$ ${\displaystyle L^{p}}$ ${\displaystyle f}$
• Are equi-integrable families, so too , for , , and equi-integrable families. The operations are always to be understood element by element, if not clear. ${\displaystyle {\mathcal {F}},{\mathcal {G}}}$ ${\displaystyle \alpha {\mathcal {F}}+\beta {\mathcal {G}}}$ ${\displaystyle \alpha ,\beta \in \mathbb {R} }$ ${\displaystyle {\mathcal {F}}\cup {\mathcal {G}}}$ ${\displaystyle {\mathcal {F}}\cap {\mathcal {G}}}$ ${\displaystyle \vert {\mathcal {F}}\vert }$

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 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• 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 .