In this wikipedia article about Radon-measures, they made the following statement:
One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous functions with compact support.
Question: Why should such a measure not have a well defined support? And why is a well defined support important to the measure we want to define on the topological space?