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196 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

where f ∈ Cc (Rp). Thus for such f , L f =∫

f dmp but mp is a complete Borel measurewhich is also regular. One dimensional Lebesgue measure has already been discussed. Iam writing dxik ≡ dm1

(xik

).

Then from Lemma 8.0.3 this functional L and Lσ give the same Borel measure mp.Here σ is the permutation which yields (i1, · · · , ip) . Now let U be an open set. Then fromTheorem 8.1.3, and letting ψn be the increasing sequence of functions in Cc (U) convergingpointwise to XU , we obtain the following from the monotone convergence theorem appliedto the indicated succession of iterated integrals,∫

∞

−∞

· · ·∫

∞

−∞

XU dxpdxp−1 · · ·dx1 = limn→∞

∫∞

−∞

· · ·∫

∞

−∞

ψndxpdxp−1 · · ·dx1

= limn→∞

∫ψndmp =

∫XU dmp

= limn→∞

∫∞

−∞

· · ·∫

∞

−∞

ψndxipdxip−1 · · ·dxi1 =∫

∞

−∞

· · ·∫

∞

−∞

XU dxipdxip−1 · · ·dxi1

This has proved part of the following result.

Lemma 8.4.2 For any E Borel and (i1, · · · , ip) a permutation,∫∞

−∞

· · ·∫

∞

−∞

XEdxipdxip−1 · · ·dxi1 =∫

XEdmp ∗

and all iterated integrals make sense. I am writing dxik for dm1(xik

).

Proof: Let S consist of the Borel sets E such that ∗ holds for E ∩ (−R,R)p. Then Scontains the open sets by what was just argued. The open sets are a π system because theyare closed with respect to finite intersections. Also, S is closed with respect to countabledisjoint unions by an application of the monotone convergence theorem on each iteratedintegral. If E ∈ S , does it follow that EC ∈ S ? First note that each iterated integral inXEC∩(−R,R)p makes sense because the corresponding integrals for XE∩(−R,R)p and X(−R,R)p

make sense and XEC∩(−R,R)p = X(−R,R)p −XE∩(−R,R)p .∫(−R,R)p

dmp =∫

∞

−∞

· · ·∫

∞

−∞

XE∩(−R,R)pdxipdxip−1 · · ·dxi1

+∫

∞

−∞

· · ·∫

∞

−∞

XEC∩(−R,R)pdxipdxip−1 · · ·dxi1

=∫

XE∩(−R,R)pdmp +∫

∞

−∞

· · ·∫

∞

−∞

XEC∩(−R,R)pdxipdxip−1 · · ·dxi1

Therefore, ∫(−R,R)p∩EC

dmp =∫

∞

−∞

· · ·∫

∞

−∞

XEC∩(−R,R)pdxipdxip−1 · · ·dxi1

and so by the lemma on π systems, S consists of the Borel sets because S contains theopen sets and the smallest σ algebra containing the open sets. Thus for any E Borel,∫

∞

−∞

· · ·∫

∞

−∞

X(−R,R)p∩Edxipdxip−1 · · ·dxi1 =∫

X(−R,R)p∩Edmp

Let R→∞ and use the monotone convergence theorem as needed to obtain that the iteratedintegrals all make sense and that the equality is preserved with E in place of (−R,R)p∩E.■

196 CHAPTER 8. POSITIVE LINEAR FUNCTIONALSwhere f € C,(R”). Thus for such f, Lf = f fdmp but mp is a complete Borel measurewhich is also regular. One dimensional Lebesgue measure has already been discussed. Iam writing dx;, = dm, (xi, )-Then from Lemma 8.0.3 this functional L and Lg give the same Borel measure my.Here o is the permutation which yields (i),--- ,i,). Now let U be an open set. Then fromTheorem 8.1.3, and letting y,, be the increasing sequence of functions in C, (U) convergingpointwise to 2y, we obtain the following from the monotone convergence theorem appliedto the indicated succession of iterated integrals,/ of Fipdxpdp dx = hirn | of Wd pp «dxnoo= lim / w,dip = / Xydmyn—yoo= lim | see /. Wdxi,dXi,_, oe -dX;, = /. oe f Ry AXxj,dXi, oe -dX;,This has proved part of the following result.Lemma 8.4.2 For any E Borel and (ij,+++ ,ip) a permutation,/ of LAX), AX, _, dX; = | %am, *and all iterated integrals make sense. I am writing dx;, for dm, (xi, ) .Proof: Let .Y consist of the Borel sets E such that * holds for EM (—R,R)?. Then 7contains the open sets by what was just argued. The open sets are a 7 system because theyare closed with respect to finite intersections. Also, .Y is closed with respect to countabledisjoint unions by an application of the monotone convergence theorem on each iteratedintegral. If E € .Y, does it follow that E© € .Y? First note that each iterated integral inBy ECn(—R,r)? Makes sense because the corresponding integrals for Le—rpy and 2(_p pymake sense and LECH —RR)? = 2 (RR)? _ ZE\(-RR)| cu dinp = [. _ [. LEn(—RRPAXipAXi,_; - -dxj,+f of Aeco(—RypyPAXipAXiy + dX),= [ Fen -ewvdmy + | vf Bncr(—RypyPAXipAXiy_ dX;Therefore,| cwpnwe dmp = [. a /. Beco(—RypyPAXipAXiy_ . -dXx;,and so by the lemma on 7 systems, . consists of the Borel sets because .Y contains theopen sets and the smallest o algebra containing the open sets. Thus for any E Borel,[ vf Zr py cdxipAXi,_, »+ dX), => [ 2% weypoedmyLet R — co and use the monotone convergence theorem as needed to obtain that the iteratedintegrals all make sense and that the equality is preserved with E in place of (—R,R)? NE.|