The measure m is called inner regular or tight if, for any open set U , m U is the supremum of m K over all compact subsets K of U. The measure m is called outer regular if, for any Borel set B , m B is the infimum of m U over all open sets U containing B. The measure m is called locally finite if every point of X has a neighborhood U for which m U is finite.
If m is locally finite, then it follows that m is finite on compact sets, and for locally compact Hausdorff spaces, the converse holds, too. Thus, in this case, local finiteness may be equivalently replaced by finiteness on compact subsets. The measure m is called a Radon measure if it is inner regular, outer regular and locally finite.
Nonlinear structures determined by measures on Banach spaces
It is possible to extend the theory of Radon measures to non-Hausdorff spaces, essentially by replacing the word "compact" by "closed compact" everywhere. However, there seem to be almost no applications of this extension. When the underlying measure space is a locally compact topological space, the definition of a Radon measure can be expressed in terms of continuous linear functionals on the space of continuous functions with compact support.
This makes it possible to develop measure and integration in terms of functional analysis , an approach taken by Bourbaki and a number of other authors. In what follows X denotes a locally compact topological space.
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Continuity with respect to the direct limit topology defined above is equivalent to the following condition: for every compact subset K of X there exists a constant M K such that, for every continuous real-valued function f on X with support contained in K ,. These real-valued Radon measures need not be signed measures. For example, sin x d x is a real-valued Radon measure, but is not even an extended signed measure as it cannot be written as the difference of two measures at least one of which is finite.
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In this set-up it is common to use a terminology in which Radon measures in the above sense are called positive measures and real-valued Radon measures as above are called real measures. To complete the buildup of measure theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure integral from compactly supported continuous functions. This can be done for real or complex-valued functions in several steps as follows:.
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