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There are two kinds of vectorvalued measures which are involved in the theory of weakly stationary processes: orthogonal (Hilbert space valued) and multiplicative (projection-valued) measures. For both classes we show that weak convergence is equivalent with the convergence of integrals over bounded continuous functions. Moreover we prove continuity theorems for the Fourier transformation as well as for the Laplace transformation of such measures.

Denote by C the Banach space of all continuous, 2-w-periodic, real-valued functions defined on [T, ni], the norm being given by the equation 11f 11 = max If(x)I . Denote by Pn the subspace of C consisting of all trigonometric polynomials of degree Pk, where po, . . ., P2n is any basis for Pn, which is orthonormal with respect to the inner-product = f' , fg. It was proved by Lozinski [6] that the Fourier projection is a minimal projection of C onto Pn. In other words, the inequality 11Fn 11 -< 11A 11 is valid for every projection A of C onto Pn. This property of Fn has been discussed by other authors [3, p. 254], [4, p. 154], and [1], and it has been an open question whether any other minimal projection from C onto Pn exists. It is proved below that Fn is the only such projection. The following conventions are used throughout the paper. An integral written simply ff or ff(t) dt stands for (1/2wT) fLn f(t) dt. For each point x e[-IT,T], an evaluation functional x is defined by the equation xf=f(x), f being an arbitrary member of C. The shift operator TJ is defined by xT8f=f(x+s). Several elementary results from Fourier analysis are required. If A is any projection of C onto Pn then Fn = f T_,AT, ds. This means that for each x and for each f, xFnf= f XT-,AT8fds. This equation of Berman [1] yields at once the inequality $ AFn