This entry should not be mistaken with the entry on the Riesz representation theorem of bounded linear functionals on an Hilbert space.

The Riesz representation theorem(s) provided here basically state that linear functionals on certain spaces of functions can be seen as integration against measures. In other words, for some spaces of functions all linear functionals have the form

There are many versions of these Riesz representation theorems, and which version is used depends upon the generality one wishes to achieve, the difficulty of proof, the type of space of functions involved, the type of linear functionals involved, the type of the "base" space involved, and also the type of measures involved.

We present here some possible Riesz representation theorems of general use.

Notation - In the following we adopt the following conventions:

• is a locally compact Hausdorff space.
• denotes the space of real valued continuous functions on with compact support.
• denotes the space of real valued continuous functions on that vanish at infinity.
• all function spaces are endowed with the sup-norm
• a linear functional is said to be positive if whenever .

Theorem 1 (Riesz-Markov) - Let be a positive linear functional on . There exists a unique Radon measure on , whose underlying -algebra is the -algebra generated by all compact sets, such that

Notice that when is -compact the underlying -algebra for these measures is precisely the Borel -algebra of .

Theorem 2 - Let be a positive linear functional on . There exists a unique finite Radon measure on such that

Theorem 3 (Dual of ) - Let be a bounded linear functional on . There exists a unique finite signed Borel measure on such that

Complex version:

Here denotes the space of complex valued continuous functions on that vanish at infinity.

Theorem 4 - Let be a bounded linear functional on . There exists a unique finite complex regular Borel measure on such that