A locally integrable function is a function that can be integrated into every compact , but this function does not have to be integratable into certain open sets. Such functions are used as aids in analysis or functional analysis . So they play an important role especially in distribution theory . In addition, the concept of locally integrable functions can be transferred to locally p-integrable functions and to locally weakly differentiable functions.
The set of these functions is denoted by. If one identifies all functions from one another, which are almost the same everywhere , one obtains the space . The equivalent definition can also be found in
connection with distribution theory
,
where is the set of equivalence classes of the measurable functions, which are almost the same everywhere, and the space of the test functions .
Instead of demanding that it is open, other authors also assume that it is compact . For the definition of space it is sufficient to assume a measurable amount . However , this generality would be unfavorable for the definition of the space of locally integrable functions, since there are measurable sets that contain no compact terms apart from zero sets . This would mean that every measurable function could be locally integrated. In addition, all semi-norms would be constant zero, so the topology they induce would be indiscreet . Functions cannot be separated in such a space . Such a pathological example is obtained with the irrational numbers .
Examples
The constant one function can be integrated locally on unlimited , but not Lebesgue integrable .
is defined for a fixed, locally integrable function . Hence, one identifies the space with the set of regular distributions based on it . With the mapping you get a continuous embedding
in the room of the distributions.
A function is generally not a member of . However, applies to everyone .
for true
.
This does not apply to the -spaces in general, unless has finite measure.
The spaces of the weakly differentiable functions are the Sobolev spaces . Since these sub-spaces of the are, it is defined areas Sobolev for these analogous local. Be open and . A function is in space if its -th weak derivative exists. This definition is equivalent to
,
where is the space of distributions . This type of Sobolev room is also a Fréchet room . For the Sobolev space corresponds to the space of the locally Lipschitz continuous functions . Is limited to one, wherein the dimension of the surrounding is, it is almost everywhere differentiable , and the gradient of agrees with the gradient in the sense of weak discharge the same. Since the space of locally Lipschitz continuous functions follows Rademacher's theorem as a special case.
Individual evidence
^ Otto Forster : Analysis. Volume 3: Measure and integration theory, integral theorems in R n and applications , 8th improved edition. Springer Spectrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5 , page 58
^ Herbert Amann, Joachim Escher : Analysis III . 1st edition. Birkhäuser-Verlag, Basel / Boston / Berlin 2001, ISBN 3-7643-6613-3 , page 129
↑ Juha Heinonen: Lectures on analysis on metric spaces . Springer, 2001, ISBN 0387951040 , pages 14-15
↑ Alain Grigis & Johannes Sjöstrand: Microlocal analysis for differential operators: an introduction , Cambridge University Press, 1994, ISBN 0-521-44986-3 , page 44
↑ Lawrence Evans: Partial Differential Equations . American Mathematical Society, ISBN 0-8218-0772-2 , pages 280-281