M. Riesz extension theorem

Remarks

The proof extends $\varphi$ one dimension at a time, as in the proof of the Hahn–Banach theorem. To add a vector $y \notin F$, one needs a value $c = \psi(y)$ with $\sup \varphi(x) \le c \le \inf \varphi(x')$, where $x$ runs over $F$ with $y - x \in K$ and $x'$ runs over $F$ with $x' - y \in K$. The hypothesis on $F$ makes both sets nonempty, and positivity of $\varphi$ on $F \cap K$ makes the supremum at most the infimum. Zorn's lemma then gives an extension to all of $E$. In the moment problem, $E$ is a space of functions containing the polynomials and $K$ is the cone of nonnegative functions in $E$. There $F$ is the space of polynomials and $\varphi$ is the moment functional $x^n \mapsto m_n$. The positive extension $\psi$ is then turned into a measure by a representation theorem, which solves the Hamburger moment problem for a positive moment functional. Without the assumption that $y - x \in K$ for some $x \in F$, an extension may fail to exist. For example, take $E = \R^2$, $K = \set{(s, t) : t > 0} \cup \set{(s, 0) : s \ge 0}$, $F = \R \times \set{0}$ and $\varphi(s, 0) = s$; then $\varphi \ge 0$ on $F \cap K$, but a positive extension $\psi(s, t) = s + ct$ would need $s + c \ge 0$ for all $s$, which is impossible.

Going deeper

For a student

The theorem is the abstract tool behind the existence parts of moment problems. A sequence $(m_n)$ defines a linear form $L$ on polynomials by $L(x^n) = m_n$. Suppose that $L(p) \ge 0$ for every polynomial $p \ge 0$. Then the theorem extends $L$ to a positive linear form on a larger space of functions, such as the continuous functions bounded by a polynomial. The Riesz representation theorem then gives a measure with moments $m_n$. The same argument, with polynomials in several variables and nonnegativity on a closed set, gives the multivariate form of this existence result.