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.
