Function that can be integrated locally

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.

definition

In this section the locally integrable function and the function space are defined. Let be an open subset and a Lebesgue measurable function . The function is said to be locally integrable if the Lebesgue integral is finite for every compact, i.e. ${\displaystyle L_{\mathrm {loc} }^{1}}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon \Omega \to \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle K\subset \Omega }$

${\displaystyle \int _{K}|f(x)|\,\mathrm {d} x<\infty }$.

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 ${\displaystyle {\mathcal {L}}_{\operatorname {loc} }^{1}(\Omega )}$ ${\displaystyle {\mathcal {L}}_{\operatorname {loc} }^{1}(\Omega )}$ ${\displaystyle L_{\operatorname {loc} }^{1}(\Omega )}$

${\displaystyle L_{\operatorname {loc} }^{1}(\Omega ):=\left\{f\in L^{0}(\Omega )\,\left|\,\int _{\mathbb {R} ^{n}}f(x)\phi (x)\,\mathrm {d} x<\infty ,\ \phi \in {\mathcal {D}}(\Omega )\right.\right\}}$,

where is the set of equivalence classes of the measurable functions, which are almost the same everywhere, and the space of the test functions . ${\displaystyle L^{0}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )\cong C_{c}^{\infty }(\Omega )}$

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 . ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \sigma }$ ${\displaystyle L^{1}\left(\Omega \right)}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle L_{\mathrm {loc} }^{1}\left(\Omega \right)}$ ${\displaystyle \left\|\,\cdot \right\|_{L^{1}\left(K\right)}\ \left(K\subset \subset \Omega \right)}$ ${\displaystyle \Omega =\mathbb {R} \setminus \mathbb {Q} }$

Examples

• The constant one function can be integrated locally on unlimited , but not Lebesgue integrable . ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$
• All functions can also be integrated locally. ${\displaystyle L^{p}}$
• The function

${\displaystyle f(x)={\begin{cases}{\frac {1}{x}}&x\neq 0\\0&x=0\end{cases}}}$

can not be integrated locally with. ${\displaystyle x=0}$

Locally p-integrable function

Analogous to the functions, functions can also be defined. Be open or compact. A measurable function is called locally p-integrable, if the expression ${\displaystyle L_{\mathrm {loc} }^{1}(\Omega )}$ ${\displaystyle L_{\mathrm {loc} }^{p}(\Omega )}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle \sigma }$ ${\displaystyle f\colon \Omega \to \mathbb {C} }$

${\displaystyle \int _{K}|f(x)|^{p}\,\mathrm {d} x\,}$

for and for all compacts exists. ${\displaystyle p\geq 1}$ ${\displaystyle K\subset \Omega }$

properties

• A regular distribution is a continuous and linear functional that passes through

${\displaystyle \phi \in {\mathcal {D}}(\mathbb {R} ^{n})\mapsto \int _{\mathbb {R} ^{n}}f(x)\phi (x)\,\mathrm {d} x}$

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 ${\displaystyle f\in L_{\mathrm {loc} }^{1}(\mathbb {R} ^{n})}$ ${\displaystyle L_{\mathrm {loc} }^{1}(\mathbb {R} ^{n})}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \textstyle f\in L_{\mathrm {loc} }^{1}(\mathbb {R} ^{n})\mapsto \left(\phi \in {\mathcal {D}}(\mathbb {R} ^{n})\mapsto \int _{\mathbb {R} ^{n}}f(x)\phi (x)\,\mathrm {d} x\right)}$

${\displaystyle L_{\mathrm {loc} }^{1}(\mathbb {R} ^{n})\hookrightarrow {\mathcal {D}}'(\mathbb {R} ^{n})}$

in the room of the distributions.

• A function is generally not a member of . However, applies to everyone . ${\displaystyle f\in L_{\mathrm {loc} }^{p}(\Omega )}$ ${\displaystyle L^{p}(\Omega )}$ ${\displaystyle L^{p}(\Omega )\subset L_{\mathrm {loc} }^{p}(\Omega )}$ ${\displaystyle 1\leq p\leq \infty }$
• for true ${\displaystyle 1\leq p<r\leq \infty }$

${\displaystyle L_{\mathrm {loc} }^{r}(\Omega )\subset L_{\mathrm {loc} }^{p}(\Omega )}$.

This does not apply to the -spaces in general, unless has finite measure. ${\displaystyle L^{p}(\Omega )}$ ${\displaystyle \Omega }$

• Let be any sequence of open, relatively compact subsets of with , then a sequence of semi-norms is on . With this semi-norm it becomes a metrizable locally convex vector space . Since all Cauchy sequences converge with regard to this metric, i.e. the space is complete , it is a Fréchet space . ${\displaystyle (\Omega _{i})_{i\in \mathbb {N} }}$ ${\displaystyle \Omega }$ ${\displaystyle \textstyle \Omega =\bigcup _{i\in \mathbb {N} }\Omega _{i}}$ ${\displaystyle \|\cdot \|_{L^{p}(\Omega _{i})}}$ ${\displaystyle L_{\mathrm {loc} }^{p}(\Omega )}$ ${\displaystyle L_{\mathrm {loc} }^{p}(\Omega )}$

Locally poorly differentiable functions

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 ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle L^{p}(\Omega )}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle f\in L_{\mathrm {loc} }^{p}(\Omega )}$ ${\displaystyle W_{\mathrm {loc} }^{k,p}(\Omega )}$ ${\displaystyle k}$

${\displaystyle W_{\mathrm {loc} }^{k,p}(\Omega ):=\left\{u\in {\mathcal {D}}'(\Omega )\mid \phi u\in W^{k,p}(\mathbb {R} ^{n}),\forall \phi \in {\mathcal {D}}(\Omega )\right\}}$,

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. ${\displaystyle {\mathcal {D}}'(\Omega )}$ ${\displaystyle p=\infty }$ ${\displaystyle W_{\mathrm {loc} }^{1,\infty }(\Omega )}$ ${\displaystyle p}$ ${\displaystyle n<p\leq \infty }$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\in W_{\mathrm {loc} }^{1,p}}$ ${\displaystyle \Omega }$ ${\displaystyle f}$ ${\displaystyle W_{\mathrm {loc} }^{1,\infty }(\Omega )}$

Individual evidence

1. ^ 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
2. ^ Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 , page 281
3. ↑ Juha Heinonen: Lectures on analysis on metric spaces . Springer, 2001, ISBN 0387951040 , page 5
4. ^ ^{A b} Elliott H. Lieb & Michael Loss: Analysis . American Mathematical Society, Second Edition, 2001, ISBN 0-8218-2783-9 , page 137
5. ^ Herbert Amann, Joachim Escher : Analysis III . 1st edition. Birkhäuser-Verlag, Basel / Boston / Berlin 2001, ISBN 3-7643-6613-3 , page 129
6. ↑ Juha Heinonen: Lectures on analysis on metric spaces . Springer, 2001, ISBN 0387951040 , pages 14-15
7. ↑ Alain Grigis & Johannes Sjöstrand: Microlocal analysis for differential operators: an introduction , Cambridge University Press, 1994, ISBN 0-521-44986-3 , page 44
8. ↑ Lawrence Evans: Partial Differential Equations . American Mathematical Society, ISBN 0-8218-0772-2 , pages 280-281

Web links

• Mathworld: Locally Integrable