Locally integrable function

A locally integrable function is a function that is integrable on any compact set, but this function on some open sets need not be integrable. Such functions are used in analysis or functional analysis as a tool. Thus, these play an important role especially in the distribution theory. In addition, you can transfer the concept of locally integrable functions on the locally p- integrable functions and to the locally weakly differentiable functions.

Definition

In this section, the locally integrable function and the function space can be defined. Be an open subset and a Lebesgue measurable function. The function is called locally integrable if for every compact the Lebesgue integral

Is finite. The amount of these functions is denoted by. If we identify all the functions from each other, which are equal almost everywhere, we obtain the space. In connection with the distribution theory to find the equivalent definition

Wherein the amount of the equivalence classes of the measurable features that are the same everywhere, and the space of the test functions.

Rather than demand that is open, it is assumed by other authors as well as compact. For the definition of the space it would be sufficient if a measurable amount would be. For the definition of locally integrable functions this is not enough, because there are measurable quantities that do not contain compact set except the empty set. This would cause any measurable function would be locally integrable.

Examples

• The constant one function is locally integrable but not Lebesgue integrable.
• All functions are locally integrable.
• The function

Local p- integrable function

Similar to the functions you can also define functions. Be open or compact. A measurable function is called locally p- integrable, if the expression

Exists for and for all compacta.

Properties

• A regular distribution is a continuous and linear functional, by the

• Is a function in general is not an element of. However, for all.
• For valid

• Be an arbitrary sequence of open relatively compact subsets of with, then a sequence of semi-norms. With this semi-norm is a metrizable locally convex vector space. Since with respect to this metric converge all Cauchy sequences, so the space is complete, it is a Fréchet space.

Local weakly differentiable functions

The spaces of weakly differentiable functions are the Sobolev spaces. Since these subspaces are the, we also defined for this quite analogous local Sobolev spaces. Be open and. A function is located in the space when the low -th derivative exists. This definition is equivalent to

The space of distributions. This type of Sobolev spaces is also a Fréchet space. For the Sobolev space is the space of locally Lipschitz continuous functions. Limited to, wherein the dimension of the surrounding is, it is almost everywhere differentiable in and coincides with the gradient of the gradient of the weak discharge in the sense of the same. Since the space of locally Lipschitz continuous functions, it follows that the set of Rademacher as a special case.