# Riesz Representation Theorem – 1

This paragraph we follow the book of Rudin [1]. On linear functionals, there is special relationship between integration and linear functionals. In ${L^1(\mu)}$, a vector space, for any positive measure ${\mu}$, the mapping

$\displaystyle f \mapsto \int_X f d\mu.$

example, let ${C([0,1])}$ be the set of all continuous functions on the unit interval ${I = [0,1]}$. Then

$\displaystyle \Lambda f = \int_0^1 f(x)dx \quad \ (f \in C([0,1])),$

has two properties:

• ${\Lambda(f + g) = \int_0^1[f(x) + g(x)]dx = \int_0^1f(x)dx + \int_0^1g(x)dx = \Lambda(f) + \Lambda(g)}$.
• ${\Lambda(c.f) =\int_0^1cf(x)dx = c\int_0^1f(x)dx= c.\Lambda(f)}$.

so it is a linear functional on ${C([0,1])}$. Moreover, this is a positive linear functional: if ${f \ge 0}$ then ${\Lambda(f) \ge 0}$.

Consider a segment ${(a,b) \subset I}$ and the class of all ${f \in C(I)}$ with ${0 \le f(x) \le 1, \forall x \in I}$ and ${f(x) = 0, \forall x \notin (a,b)}$ or support of ${f}$ is ${(a,b) \subset I}$. So we get ${\Lambda(f) = \int_a^b f(x)dx \le \int_a^bdx = b-a}$. It is important that the length of ${(a,b)}$ related to the values of the functional ${\Lambda}$.

There is an important theorem of F. Riesz, this illustrates to above event

Theorem 1 (F. Riesz) Let $X$ be a closed subset of $\mathbb{R}$. Then every positive linear functional ${\Lambda}$ on ${C(X)}$ there corresponds a finite positive Borel measure ${\mu}$ on ${X}$ such that

$\displaystyle \Lambda(f) = \int_X fd\mu \quad \ (f \in C(X)).$

References

[1] W. Rudin, Real and Complex analysis, Mc.Graw-Hill, 1970.