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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


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

© 1996-9 Eric W. Weisstein