Riesz representation theorem

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

A central theorem in classical measure theory, sometimes called Riesz-Markov theorem, which states the following. Let $X$ be a compact Hausdorff topological space, $C(X)$ the Banach space of real valued continuous functions on $X$ and $L: C(X)\to \mathbb R$ a continuous linear functional which is nonnegative, i.e. such that $L(f)\geq 0$ whenever $f\geq 0$. Then there is a Radon measure $\mu$ on the $\sigma$-algebra of Borel sets $\mathcal{B} (X)$ such that \[ L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, . \] The theorem is often stated in a more general version for locally compact Hausdorff spaces $X$, of which the statement above is a simple corollary (cp, with Section 2.14 of [Ru])

An analogous statement which is commonly referred to as Riesz representation theorem is that, under the assumptions above, the dual of $C(X)$ is the space $\mathcal{M}^b (X)$ of $\mathbb R$-valued measures with finite total variation (cp. with Signed measure for the relevant definitions). Combined with the Radon-Nikodým theorem, this amounts to the following alternative statement: for any element $L\in (C(X))'$ there are a Radon measure $\mu$ and a Borel function $g$ such that $|g|=1$ $\mu$-a.e. and \[ L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, . \]

The statement can be also generalized to a similar description of the dual of $C(X,B)$ when $B$ is Banach space. For instance, if $B$ is a finite-dimensional space, then for any $L\in C(X,B)'$ there is a Radon measure $\mu$ on $X$ and a Borel measurable map $g: X\to B'$ such that $\|g\|_{B'}=1$ $\mu$-a.e. and \[ L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X, B)\, . \]

References

[AmFuPa]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Bo]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Bi]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ru]	W. Rudin, "Real and complex analysis". McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966 MR021052 Zbl 0142.01701