What is the support of a Radon measure on a locally compact Hausdorff space?

MHB
Thread starter Euge
Start date Apr 10, 2019

In summary, a Radon measure is a type of measure used in mathematical analysis to describe the size or magnitude of a set or function. Its support is the smallest closed set where the measure is non-zero, and it is important for understanding the behavior and properties of the measure. The support is closely related to the concept of compactness, and it can be empty if the measure is defined on a space without non-zero elements.

Apr 10, 2019

Euge

Gold Member

MHB

POTW Director

Here is this week's POTW:

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Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Recall that the complement of the support of $\mu$ is the union of all open subsets of $X$ of $\mu$-measure zero. Show that the support of $\mu$ is the set of all $x\in X$ such that for all compactly supported continuous functions $f : X\to [0,1]$ with $f(x) > 0$, the integral $\int_X f\, d\mu > 0$.-----

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What is the support of a Radon measure?

The support of a Radon measure on a locally compact Hausdorff space is the smallest closed subset of the space that has full measure. In other words, it is the set of points where the measure is non-zero.

Why is the support important in Radon measures?

The support of a Radon measure is important because it tells us where the measure is concentrated. It can also give us information about the underlying structure of the space, such as the presence of singularities or discontinuities.

Can the support of a Radon measure be empty?

Yes, the support of a Radon measure can be empty if the measure assigns zero measure to all subsets of the space. This can happen, for example, if the space is infinite and the measure is finite.

How is the support of a Radon measure related to compactness?

The support of a Radon measure is a closed subset of the space, and it is compact if and only if the space is compact. This means that the support can give us information about the compactness of the space.

What is the difference between the support of a Radon measure and the support of a function?

The support of a Radon measure is a subset of the space, while the support of a function is a subset of the domain of the function. Additionally, the support of a Radon measure is always closed, while the support of a function may not be. Finally, the support of a Radon measure can have full measure, while the support of a function is always a subset of the zero set of the function.