The Riesz Representation Theorem – I
Finally I figured out how to convert LaTeX to WordPress content. Thanks to LaTeX to WordPress. My first post is an experimental one. I have chosen Riesz representation theorem (in abstract integration). I have tried to keep the exposition simple. Any question/comment/criticism/suggestion is invited.
In this article we shall try to understand the Riesz representation theorem in measure theory. Anyone who is acquainted with Hilbert space basics knows that the Riesz representation theorem associates each linear functional to an inner product. Similarly, in the study of abstract integration the theorem associates to each positive linear functional a unique measure. The exact statement will appear later. This helps in establishing the Lebesgue measure on .
To begin with, we give some definitions and results that we shall be using. We shall be following the material and proof given in Rudin’s “Real and Complex Analysis” [Theorem 2.14], and this article is meant to fill the missing details.
represents a topological space. is Hausdorff, if any two distinct points in can be separated by disjoint open sets in . is locally compact if every point has a neighborhood (an open set containing ) whose closure is compact, i.e., is compact.
Let be a complex function on . Let The support of is then and is denoted by . The collection of all continuous complex functions on whose support is compact is denoted by . It is easy to verify that is a complex vector space.
The notation means that is a compact subset of ; ; and for all . Hence The notation means that is open; ; and . The notation means both the above statements hold true. Note that implies .
We shall use a major result – the Urysohn’s lemma. It is basically a statement about separating subsets by continuous function. Here we shall be using the following form. [Theorem 2.12]
Theorem 1. Suppose is a locally compact Hausdorff space, is open in , , and K is compact. Then there exists an , such that
Interpreted in other way, we get the existence of a continuous function with compact support which vanishes outside and assumes 1 on . Moreover, . Using Urysohn’s lemma we get the existence of a partition of unity on .
Theorem 2. Suppose are open subsets of a locally compact Hausdorff space , is compact, and Then there exist functions such that
The collection is called a partition of unity on subordinate to the cover .
A linear functional is positive if then
We now state the Riesz representation theorem.
Theorem 3. Let be a locally compact Hausdorff space, and let be a positive linear functional on . The there exists a algebra in which contains all Borel sets in , and there exists a unique positive measure on which represents in the sense that
(a). for every
and which has the additional properties:
(b) for every compact set .
(c) For every , we have
(d) For every open set we have This also holds true if with .
(e) If , , and , then .
We shall prove this theorem in the next post.