Theory of functions of a real variable

axioms for a space on which we can define integration. Furthermore, the set of such functions is an algebra containing 1 and which separates points, so is dense in C(Ω) by the Stone-Weierstrass theorem. Let us call the space of such functions Cfin(Ω). If we define an integral I on Cfin(Ω) then, by the Stone-Weierstrass theorem it extends to C(Ω) and therefore, by the Riesz representation theorem, gives us a regular Borel measure on Ω. For each x ∈ R we are going to define such an integral, Ix by Ix(φ) = ∫ · · · ∫ F (x1, x2, . . . , xm)p(x, x1; t1)p(x1, x2; t2−t1) · · · p(xm−1, xm, tm−tm−1)dx1 . . . dxm when φ = φF,t1,...,tm where p(x, y; t) = 1 (2πt)n/2 e−(x−y) 2/2t (7.2) (with p(x,∞) = 0) and all integrations are over Ṙ. In order to check that this is well defined, we have to verify that if F does not depend on a given xi then we get the same answer if we define φ in terms of the corresponding function of the remaining m− 1 variables. This amounts to the computation ∫ p(x, y; s)p(y, z, t)dy = p(x, z; s+ t). If n = 1 this is the computation 1 2πt ∫ R e−(x−y) 2/2se−(y−z) 22tdy = 1 2π(s+ t) e(x−z) 2/2(s+t). If we make the change of variables u = x− y this becomes nt ? ns = nt+s where nr(x) := 1 √ r e−x 2/2r. In terms of our “scaling operator” Sa given by Saf(x) = f(ax) we can write nr = r− 1 2S r− 1 2 n where n is the unit Gaussian n(x) = e−x 2/2. Now the Fourier transform takes convolution into multiplication, satisfies (Saf )̂ = (1/a)S1/af̂ , 7.1. WIENER MEASURE. 189 and takes the unit Gaussian into the unit Gaussian. Thus upon Fourier transform, the equation nt ? ns = nt+s becomes the obvious fact that e−sξ 2/2e−tξ 2/2 = e−(s+t)ξ 2/2. The same proof (or an iterated version of the one dimensional result) applies in n-dimensions. So, for each x ∈ R we have defined a measure on Ω. We denote the measure corresponding to Ix by prx. It is a probability measure in the sense that prx(Ω) = 1. The intuitive idea behind the definition of prx is that it assigns probability prx(E) := ∫