Quasi-integrability

In mathematics, quasi-integrability is a property that measurable functions and random variables can be assigned to, and accordingly one speaks of quasi-integrable functions and quasi-integrable random variables . Thus it can be assigned to the theory of measure and stochastics . The quasi-integrability is an important step on the way from the Riemann integral to a more general integral term, the Lebesgue integral .

definition

Quasi-integrable function

Be

${\displaystyle f\colon (\Omega ,{\mathcal {A}})\to ({\overline {\mathbb {R} }},{\mathcal {B}}({\overline {\mathbb {R} }}))}$

a measurable numerical function on the dimension space as well ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$

${\displaystyle f^{+}:=\max\{f,0\}{\text{ und }}f^{-}:=-\min\{f,0\}}$

the positive or negative part of the function. Then the function is called -quasi- integrable or quasi-integrable with respect to if at least one of the two integrals ${\displaystyle \mu }$ ${\displaystyle \mu }$

${\displaystyle I^{+}:=\int _{\Omega }f^{+}\mathrm {d} \mu {\text{ oder }}I^{-}:=\int _{\Omega }f^{-}\mathrm {d} \mu }$

is finite. If it is clear which dimension is involved, this information is generally not given. ${\displaystyle \mu }$

Quasi-integrable random variable

Let be a random variable from the probability space after . Let it be like above ${\displaystyle X}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle ({\overline {\mathbb {R} }},{\mathcal {B}}({\overline {\mathbb {R} }}))}$

${\displaystyle X^{+}:=\max\{X,0\}{\text{ und }}X^{-}:=-\min\{X,0\}}$

the positive and negative parts of the random variable. The random variable is then called -quasi- integrable or quasi-integrable with respect to if at least one of the two expected values ${\displaystyle P}$ ${\displaystyle P}$

${\displaystyle \operatorname {E} (X^{+}){\text{ und }}\operatorname {E} (X^{-})}$

is finite. If it is clear which probability measure is meant, this is usually not given. ${\displaystyle P}$

Remarks

• In fact, the two definitions agree, only they are formulated in the notation of two different areas of mathematics. The only difference is that with the quasi-integrable random variable only probability measures and not arbitrary measures are allowed.
• The expected value of a random variable can be defined for quasi-integrable random variables, but can then possibly assume the value . Various versions of the statement "the expected value exists" exist in the literature. Some demand that it be finite, while others allow it to accept the values . Pay attention to the exact definition of the textbook here. ${\displaystyle \pm \infty }$ ${\displaystyle \pm \infty }$

use

Quasi-integrable functions play an important role in the construction of the Lebesgue integral .

First the integral is only defined for the class of positive simple functions and then generalized to positive measurable functions by an approximation argument. In order to define the integral for any measurable function, measurable functions are broken down into their positive and negative parts

${\displaystyle f^{+}:=\max\{f,0\}{\text{ und }}f^{-}:=-\min\{f,0\}}$

and defines the integral over the function as the sum of the positive and negative parts

${\displaystyle I(f):=I(f^{+})-I(f^{-})}$.

Now, however, the expressions and both can take the value , which leads to the undefined statement ${\displaystyle I(f^{+})}$ ${\displaystyle I(f^{-})}$ ${\displaystyle +\infty }$

${\displaystyle I(f)=+\infty -\infty }$

would lead. To avoid this, quasi-integrability is required, which guarantees that only one of the integrals is always infinite. In this sense, integrals exist over quasi-integrable functions, so they are mathematically well-defined, but can definitely assume the value or . ${\displaystyle \infty }$ ${\displaystyle -\infty }$

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 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .