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Alternative formulations

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The statement of the theorem

It is possible to develop the integral with respect to the Lebesgue measure without relying on the full machinery of measure theory. One such approach is provided by the Daniell integral. There is also an alternative approach to developing the theory of integration via methods of functional analysis. The Riemann integral exists for any continuous function f of compactsupport defined on R^{n} (or a fixed open subset). Integrals of more general functions can be built starting from these integrals. Let C_{c} be the space of all real-valued compactly supported continuous functions of R. Define a norm on C_{c} by ∥f∥=∫∣f(x)∣ dx.\left\|f\right\|=\int |f(x)|\,dx. Then C_{c} is a normed vector space (and in particular, it is a metric space.) All metric spaces have Hausdorff completions, so let L^{1} be its completion. This space is isomorphic to the space of Lebesgue integrable functions modulo the subspace of functions with integral zero. Furthermore, the Riemann integral ∫ is a uniformly continuous functional with respect to the norm on C_{c}, which is dense in L^{1}. Hence ∫ has a unique extension to all of L^{1}. This integral is precisely the Lebesgue integral. More generally, when the measure space on which the functions are defined is also a locally compacttopological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) an integral with respect to them can be defined in the same manner, starting from the integrals of continuous functions with compact support. More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure is defined as a continuous linear functional on this space. The value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Nicolas Bourbaki and a certain number of other authors. For details see Radon measures. - ^Bourbaki 2004.