There are a couple of versions of this theorem. Basically, it says that any bounded linear functional on the space of compactly supported continuous functions on is the same as integration against a measure ,
Here, the integral is the Lebesgue integral.
Because linear functionals form a vector space, and are not "positive," the measure may not be a positive measure. But if the functional is positive, in the sense that implies that , then the measure is also positive. In the generality of complex linear functionals, the measure is a complex measure. The measure is uniquely determined by and has the properties of a regular Borel measure. It must be a finite measure, which corresponds to the boundedness condition on the functional. In fact, the operator norm of , , is the total variation measure of , .
Naturally, there are some hypotheses necessary for this to make sense. The space has to be locally compact and a T2-Space, which is not a strong restriction. In fact, for unbounded spaces , the theorem also applies to functionals on continuous functions which vanish at infinity, in the sense that for any , there is a compact set such that for any not in , (which is the notion from calculus of ).
The Riesz representation theorem is useful in describing the dual vector space to any space which contains the compactly supported continuous functions as a dense subspace. Roughly speaking, a linear functional is modified, usually by convolving with a bump function, to a bounded linear functional on the compactly supported continuous functions. Then it can be realized as integration against a measure. Often the measure must be absolutely continuous, and so the dual is integration against a function.