Chapter 8

Positive Linear FunctionalsIn this chapter is a standard way to obtain many examples of measures from extendingpositive linear functionals. I will consider positive linear functionals defined on Cc (X)where (X ,d) is a metric space. This can all be generalized to X a locally compact Hausdorffspace, but I don’t have many examples which need this level of generality. Also, we willalways assume that the closure of balls in (X ,d) are compact. Thus you see that the mainexample is Rp or some closed subset of Rp like a m dimensional surface in Rp wherem < p. This approach will not work for finding measures on infinite dimensional Banachspaces for example, which appears to limit its applications to probability but it is a verygeneral approach which gives outstanding results very quickly. To see more generalityincluding the locally compact Hausdorff spaces, see Rudin [39] which is where I first sawthis, actually in an earlier version of this book. Another source is in Hewitt and Stromberg[22].

Lemma 8.0.1 Let f ∈C ([a,b]) . Then∫ b

a f dx =∫[a,b] f dm1. The Riemann integral from

calculus equals the Lebesgue integral.

Proof: One can reduce to the case where f (x) ≥ 0 by looking at positive and nega-tive parts of real and imaginary parts. Let a = x0 < · · · < xn = b and consider the stepfunctions fn (x) ≡ ∑

nk=1 f (xk−1)X[xk−1,xk) (x). This converges uniformly to f and each is

a simple function. By the fact that m1 (I) is just the length of I for I an interval, we have∫ ba fn (s)dx =

∫[a,b] fndm1. Now let n→ ∞ and use the uniform convergence to conclude∫ b

a f dx =∫[a,b] f dm1. ■

This is based on extending functionals. The most obvious functional is as follows:

L f ≡∫

∞

−∞

· · ·∫

∞

−∞

f (x1, · · · ,xp)dxpdxp−1 · · ·dx1 (8.1)

the iterated integral in which f ∈ Cc (Rp). You do exactly what the notation says. Firstintegrate with respect to xp then with respect to xp−1 and so forth. This makes perfectsense whenever f ∈Cc (Rp) and we can consider each iterated integral as either a Riemannintegral from Calculus or a Lebesgue integral with respect to dm1 since the above lemmashows these are the same.

Lemma 8.0.2 The functional L makes sense for f ∈Cc (Rp) .

Proof: Let f be zero off [−R,R]p a compact set. Then by uniform continuity of f onthis compact set, if

∣∣x̂p−1− xp−1∣∣ is small enough,∣∣ f (x1, · · · , x̂p−1,xp)− f (x1, · · · ,xp−1,xp)

∣∣< ε/2R

Therefore, for∣∣x̂p−1− xp−1

∣∣ this small,∣∣∣∣∫ ∞

−∞

f (x1, · · · ,xp−1,xp)dxp−∫

∞

−∞

f (x1, · · · , x̂p−1,xp)dxp

∣∣∣∣=

∣∣∣∣∫ R

−Rf (x1, · · · ,xp−1,xp)dxp−

∫ R

−Rf (x1, · · · , x̂p−1,xp)dxp

∣∣∣∣185