Birkhoff integral

The Birkhoff integral is an integral term that was introduced by Garrett Birkhoff in 1935 for the integration of functions that are equivalent to banach space . While the Bochner integral is the direct generalization of Lebesgue's integral concept to Banach space-valued functions, the Birkhoff integral represents a generalization of the Riemann integral in two respects . On the one hand, functions are now considered which are defined over any finite dimensional space . Furthermore, not only finite sums (the so-called Riemann sums) are considered, but also convergent series . While every Riemann-integrable function can be integrated on the Lebesgue, on the other hand it is true that every Bochner-integrable function must be Birkhoff-integrable on a -finite dimensional space. ${\displaystyle \textstyle \sigma }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \textstyle \sigma }$

definition

Let there be an infinite measure space and a Banach space and a function. In preparation for the actual definition, three basic abbreviations are introduced here: ${\displaystyle \textstyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \textstyle \sigma }$ ${\displaystyle \textstyle (B,\|\cdot \|)}$ ${\displaystyle \textstyle f\colon \Omega \to B}$

• For a set the diameter is defined by . ${\displaystyle \textstyle \emptyset \neq M\subseteq B}$ ${\displaystyle \textstyle \mathrm {diam} (M):=\sup _{x,y\in M}\|x-y\|}$
• For a set denotes the convex hull of . ${\displaystyle \textstyle \emptyset \neq M\subseteq B}$ ${\displaystyle \textstyle \mathrm {konv} (M)}$ ${\displaystyle \textstyle M}$

• A subset of algebra is called a countable partition of if ${\displaystyle \textstyle \Gamma }$ ${\displaystyle \textstyle \sigma }$ ${\displaystyle \textstyle {\mathcal {A}}}$ ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle \Omega }$
  • ${\displaystyle \textstyle \Gamma }$is a countable partition of and ${\displaystyle \textstyle \Omega }$
  • has any quantity in finite measure, so we have . ${\displaystyle \textstyle \Gamma }$ ${\displaystyle \textstyle \forall M\in \Gamma :\mu (M)<\infty }$

With the help of these terms, the Birkhoff integral can be defined as a generalization of the Riemann integral. First, the notion of Riemann sums is generalized over a partition of the domain:

${\displaystyle \textstyle f}$is called unconditionally summable under the countable partition of , if the following applies: is unconditionally convergent . ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle \Gamma }$ ${\displaystyle \textstyle \Omega }$ ${\displaystyle \textstyle \forall (b_{M})_{M\in \Gamma }\in \prod _{M\in \Gamma }f(M)\colon \sum _{M\in \Gamma }\mu (M)b_{M}}$

Every formally possible countable Riemann sum over the partition must therefore necessarily be convergent. In the next definition all Riemann sum values ​​of this partition are collected: ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle \mu }$

${\displaystyle \sum _{M\in \Gamma }\mu (M)f(M):={\bigg \{}\sum _{M\in \Gamma }\mu (M)b_{M}\,{\Big |}\,\forall \,M\in \Gamma :b_{M}\in f(M){\bigg \}}}$.

One calls (unconditionally) Birkhoff integrable if there is a sequence of countable partitions with is unconditionally summable under and also still applies ${\displaystyle \textstyle f}$ ${\displaystyle \textstyle (\Gamma _{i})_{i\in \mathbb {N} }}$ ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle \forall \,i\in \mathbb {N} :f}$ ${\displaystyle \textstyle \Gamma _{i}}$

${\displaystyle \displaystyle \lim _{i\to \infty }\mathrm {diam} {\bigg (}{\overline {\mathrm {konv} {\Big (}\sum _{M\in \Gamma _{i}}\mu (M)f(M){\Big )}}}{\bigg )}=0}$.

The diameters of the sets of Riemann sum values ​​belonging to the partition sequence (previously convex and then topologically closed) must converge to zero. Then there is exactly one element on average ${\displaystyle \textstyle b}$

${\displaystyle \bigcap _{i=1}^{\infty }{\overline {{\rm {konv}}{\Big (}\sum _{M\in \Gamma _{i}}\mu (M)f(M){\Big )}}}}$.

This is also independent of the specific choice of the sequence and is defined as the (unconditional) Birkhoff integral ${\displaystyle \textstyle (\Gamma _{i})_{i\in \mathbb {N} }}$

${\displaystyle \int _{\Omega }f\,{\rm {d}}\mu :=b}$.

Comparison with other integral terms

• Every Bochner-integrable function defined in a finite dimensional space can also be Birkhoff-integrable and the corresponding integral values ​​then match. However, there are Birkhoff-integrable functions that cannot be Bochner-integrated. ${\displaystyle \textstyle \sigma }$
• If the definition of the Riemann integral is generalized directly to Banach space-valued functions by means of Riemann sums, then in general not every Riemann-integrable function can also be Bochner-integrable, but Birkhoff-integrable instead.
• An example of a non-Bochner-integrable but Birkhoff-integrable (even Riemann-integrable) function is:

Be provided with the norm , see general space and , where the image from below is precisely the characteristic function of . ${\displaystyle \textstyle \;\ell ^{2}\left([0,1]\right):=\left\{(x_{i})_{i\in [0,1]}\in \mathbb {R} ^{[0,1]}\;{\Big |}\;\sum _{i\in [0,1]}x_{i}^{2}<\infty \right\}\,}$ ${\displaystyle \textstyle \|(x_{i})_{i\in [0,1]}\|:={\sqrt {\left(\sum _{i\in [0,1]}x_{i}^{2}\right)}}}$ ${\displaystyle \textstyle \ell ^{p}}$ ${\displaystyle \textstyle f\colon [0,1]\to \ell ^{2}\left([0,1]\right);\,x\mapsto \chi _{\{x\}}}$ ${\displaystyle \textstyle x}$ ${\displaystyle \textstyle f}$ ${\displaystyle \textstyle x}$

${\displaystyle \textstyle f}$Bochner cannot be integrated, because otherwise it would also be measurable. With the help of Pettis' measurability theorem, however, it follows that it cannot be measured, because it is not separable almost everywhere. The Riemann integral and thus also the Birkhoff integral of is . ${\displaystyle \textstyle f}$ ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle f}$ ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle f\left([0,1]\right)}$ ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle f}$ ${\displaystyle \textstyle {\mathfrak {0}}\colon [0,1]\to \mathbb {R} ;\,x\mapsto 0}$

• Every function that can be integrated by Birkhoff can be integrated in Pettis .

properties

• The Birkhoff integral is linear . For two Birkhoff-integrable functions and Birkhoff can also be integrated and the following applies: ${\displaystyle \textstyle f,g\colon \Omega \to B}$ ${\displaystyle \alpha ,\beta \in \mathbb {K} }$ ${\displaystyle \alpha f+\beta g}$

${\displaystyle \int _{\Omega }\alpha f+\beta g\,{\rm {d}}\mu =\alpha \int _{\Omega }f\,{\rm {d}}\mu +\beta \int _{\Omega }g\,{\rm {d}}\mu }$.

• There is a relatively new equivalent characterization for the Birkhoff integrability of , see M. Potyrala: ${\displaystyle \textstyle f\colon \Omega \to B}$

${\displaystyle f}$is Birkhoff-integrable with if and only applies ${\displaystyle \textstyle x=\int _{\Omega }f\,{\rm {d}}\mu }$

${\displaystyle \forall \,\varepsilon >0\,\exists \,}$a countable partition can be summed under and . ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle \Gamma :f}$ ${\displaystyle \textstyle \Gamma }$ ${\displaystyle \textstyle \;\;\sup {\Big \{}\,\|y-x\|\,{\Big |}\,y\in \sum _{M\in \Gamma }\mu (M)f(M)\,{\Big \}}<\varepsilon }$

• Let it be another Banach space, Birkhoff-integrable and a continuous linear operator . Then the chaining is a Birkhoff-integrable function and the following applies: ${\displaystyle \textstyle D}$ ${\displaystyle f\colon \Omega \to B}$ ${\displaystyle \textstyle T\in L(B,D)}$ ${\displaystyle \textstyle T\circ f\colon \Omega \to D}$

${\displaystyle T\left(\int _{\Omega }f\mathrm {d} \mu \right)=\int _{\Omega }T\circ f\mathrm {d} \mu }$.

literature

• Jürgen Friedrich: Integration of banach space-valued functions: Bochner and Birkhoff integration. Diplomica Verlag, Hamburg 2013, pp. 28–46, ISBN 978-3-8428-4043-0 .
• Garrett Birkhoff : Integration of Functions with Values ​​in a Banach Space. In: Transactions of the American Mathematical Society, Vol. 38, No. 2 (1935), pp. 357-378.
• M. Potyrala: Some Remarks about Birkhoff and Riemann-Lebesgue Integrability of Vector Valued Functions. In: Tatra Mountains Mathematical Publications, 35 (2007), pp. 97-106.