Historical note

(N.B. — The Roman numerals refer to the bibliography at the end of this note.)

With the development of the 'vector calculus' in the course of the 19th century, it was current practice to have to integrate vector-valued functions, but as long as it was only a question of functions with values in finite-dimensional spaces, this operation posed no problem. It is only with Hilbert's spectral theory that one meets operations that lead naturally to a more general concept of integral: this theory in effect leads to associating, with every continuous Hermitian form $\Phi(x, y)$ on a Hilbert space $\mathbf{H}$, a family $(E(\lambda))_{\lambda \in \mathbf{R}}$ of orthogonal projectors having the property that, for every pair $(x, y)$ of vectors of $\mathbf{H}$, the function $\lambda \mapsto (E(\lambda)x|y)$ is of bounded variation and $\Phi(x, y) = \int \lambda d(E(\lambda)x|y)$; if one associates with $\Phi$ the Hermitian operator $A$ such that $\Phi(x, y) = (Ax|y)$, it was tempting to write the preceding formula as $A = \int \lambda dE(\lambda)$. But it was only from about 1935 on, after the introduction by Bochner of the ('strong') integration of a function with values in a Banach space, that one began to be preoccupied with defining the integral of vector-valued functions (or the integral with respect to a vectorial measure) in such a way as to be able to legitimately write formulas such as the preceding one. This extension was accomplished essentially by Gelfand (III), Dunford and Pettis (IV) and (V); their results are stated for Banach spaces, but extend without difficulty to more general locally convex spaces.

The idea of decomposing a volume into 'slices' and reducing an integral over the volume to an integral over each slice, followed by a single integration, has been used in Analysis ever since the beginning of the infinitesimal Calculus (the 'Calculus of indivisibles' of Cavalieri being nothing more than a first outline of this principle, which could even be traced back to Archimedes (see the Hist. Note for Book IV, Chs. I–III)). But in the classical applications, the 'slices' were always of a very special and very regular nature (most often open subsets of analytic surfaces, depending analytically on a parameter); it could scarcely have been otherwise in the absence of a general theory of integration. The general problem of the disintegration of a measure was posed and solved by von Neumann in 1932, in connection with ergodic theory (I); at about the same time (and independently) Kolmogoroff, while laying down axiomatic foundations for the Theory of Probability, was led to define in a general way the concept of 'conditional probability' and to prove its existence, a problem essentially equivalent to that of disintegration of a measure (II).

(I) J. von NEUMANN, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math., (2), 33 (1932), pp. 587–642.

(II) A. KOLMOGOROFF, Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin (Springer), 1933.

(III) I. GELFAND, Abstrakte Funktionen und lineare Operatoren, Mat. Sborn., (N.S.), 4 (1938), pp. 235–284.

(IV) N. DUNFORD, Uniformity in linear spaces, Trans. Amer. Math. Soc., 44 (1938), pp. 305–356.

(V) N. DUNFORD AND B. PETTIS, Linear operations on summable functions, Trans. Amer. Math. Soc., 47 (1940), pp. 323–392.