Riesz's Theorem

Every continuous linear functional $U[f]$ for $f\in C[a, b]$ can be expressed as a Stieltjes Integral

$$U[f]=\int_a^b f(x)\,dw(x),$$

where $w(x)$ is determined by $U$ and is of bounded variation on $[a,b]$.

See also Stieltjes Integral

References

Kestelman, H. ``Riesz's Theorem.'' §11.5 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 265-269, 1960.