CHAPTER 1 Integration on Locally Compact Space 1. The Integral Let T be a locally compact separated space. (Recall “separated” is equivalent to “Hausdorff.”) A separated space is locally compact if every point has a compact neighborhood. Let C(T ) be the set of all continuous, complex valued functions f defined on T . Recall that the support of f C(T ) is the closure of the set x : f(x) =0 . We define ∈ { 6 } C0(T ) to be the subspace of C(T ) consisting of functions of compact support. On C0(T ) we can define the norm f = sup f(x) : x T . r + k k {| | r∈ +} We introduce further the spaces C0 (T ), C0 (T ) or more briefly C0 , C0 : r (1) C0 (T ) is all real valued functions in C0(T ). + (2) C0 (T ) is all nonnegative functions in C0(T ). Definition 1.1. An integral I is a linear functional on + C0(T ) such that I(f) 0 whenever f C (T ). ≥ ∈ 0 r It follows that I(f) is real if f C0 (because f can be written as + − ± + ∈ f = f f where f C0 ). Also if f g then I(f) I(g) for any f,g C−r. ∈ ≥ ≥ Example∈ : The Riemann integral defined on R is an example of an integral because every function f C0(R) is Riemann integrable and of course a nonnegative function has∈ a nonnegative integral. Lemma 1.2. I(f) I( f ), for all f C0. | |≤ | | ∈ Proof. As a complex number I(f)= reiθ where r = I(f) . If we expand e−iθf = u + iv where u,v Cr then we have | | ∈ 0 r = e−iθI(f)= I(e−iθf) = I(u + iv) = I(u)+ iI(v) = I(u) I( u ) I( u + iv )= I( f ) ≤ | | ≤ | | | | which is precisely I(f) I( f ) . | |≤ | | Our objective is to extend the integral to a wider class of functions. The extension procedure below, if applied to the Riemann integral, would lead to the Lebesgue integral. 1 2 1. INTEGRATION ON LOCALLY COMPACT SPACE Lemma 1.3. If Q is a compact subset of T then there is a con- stant C > 0 so that for all f C0(Q) Q ∈ I(f) C f | |≤ Qk k Proof. Urysohn’s Lemma applies (Theorem ??) and allows us to + conclude that there exists a function g C0 (T ) so that g(x)=1if x Q. (It was shown that a compact∈ Hausdorff space is normal in the∈ Example prior Theorem ?? so that Urysohn’s Lemma applies in a compact neighborhood of Q.) Therefore for all f C0(Q) ∈ I(f)= I(gf) I( f g)= f I(g) C f ≤ k k k k ≡ Qk k where CQ = I(g). + Proposition 1.4. Suppose that X C0 (T ) is directed down- wards (that is for every f,g X there is ⊆h X so that h f and h g). Suppose further that,∈ for each t T ∈ ≤ ≤ ∈ inf f(t)=0 f∈X Then, for all ǫ> 0, there is f X so that f <ǫ and therefore ǫ ∈ k ǫk inf I(f)=0 f∈X Proof. For ǫ > 0, and f X define A (f) = t : f(t) ǫ . If ∈ ǫ { ≥ } f g then A (f) A (g). Because inf ∈ f(t)=0 ∈ A (f) = . ≥ ǫ ⊇ ǫ f X ∩f X ǫ ∅ For any f X, A (f) is compact and so there must be finitely many f1, ∈ ǫ f2,..., f so that 1≤ ≤ A (f )= . Because X is directed downward, n ∩ k n ǫ k ∅ there is fǫ X so that Aǫ(fǫ) = or in other words fǫ(t) < ǫ. (Indeed given∈ f X we can choose∅ f X so that f | f and| so ∈ ǫ ∈ ǫ ≤ inff∈X I(f) = 0, by the preceding Lemma. Semi-continuous Functions The integral extends to “lower semi- continuous” functions, we shall see but we pause to introduce the con- cept of semi-continuity and some of its elementary properties. Definition An extended valued function f : T [ , ] is said → −∞ ∞ to be lower semi-continuous (resp. upper semicontinuous) at t0 T if ∈ for each real h<f(t0) (resp. h>f(t0)) there is a neighborhood U(t0) of t0 so that f(t) > h (resp. f(t) < h) for all t U(t0). If f is lower semi-continuous (resp. upper semi-continuous) at∈ every point then it is said to be lower semi-continuous (resp. upper semi-continuous). It is clear that f is lower semi-continuous if and only if f is upper semi-continuous. It is also clear that a finite valued function− is contin- uous at t0 if and only if it is both upper and lower semi-continuous. (By contrast Royden defines lower semi-continuity at t0 further requires that f(t0) > whereas, in the present context, f(t0)= implies −∞ −∞ 1.THEINTEGRAL 3 lower semi-continuity at t0.) This definition makes sense if T is any topological space. Exercise: In the special case that T = Rn, show that a function n f : R [ , ] is lower semi-continuous at t0 if and only if → −∞ ∞ lim inf f(t) f(t0) t→t0 ≥ Consequently for functions of a real variable (n = 1) with a jump discontinuity at t0, f(t0) limt→t0+ f(t) and f(t0) limt→t0− f(t) both. ≤ ≤ Let M denote the set of all lower semi-continuous functions on T and M + be all the nonnegative functions in M. Then (1) If α> 0 then f M implies αf M. (2) If f,g M then∈min f,g M.∈ (3) If f,g ∈ M and f + g{is defined}∈ everywhere (recall +( ) is undefined)∈ then f + g M. ∞ −∞ (4) If X M then sup f : f∈ X M. (5) If X ⊆ M + then { f ∈ M}∈+. ⊆ f∈X ∈ The verification of PartP 1 is straightforward. As for Part 2, suppose that min f,g (t0) >h so that f(t0) >h and g(t0) >h and so there are { } neighborhood U(t0) and V (t0) of t0 so that f(t) >h for t U(t0) and ∈ g(t) >h for t V (t0). Consequently min f,g (t) >h on U(t0) V (t0). ∈ { } ∩ Consider next f + g. If f(t0) + g(t0) > h then there exists h1 and h2 = h h1 so that f(t0) > h1 and g(t0) > h2. There is a − neighborhood U(t0) of t0 so that f(t) > h1 if t U(t0) and there is ∈ another neighborhood V (t0) of t0 so that g(t) >h2 if t V (t0) and so ∈ on U(t0) V (t0), (f + g)(t) >h1 + h2 = h which verifies Part 3. ∩ Suppose that, for some h R, and t0 T , sup f(t0): f X >h. ∈ ∈ { ∈ } Therefore there must be f0 X so that f0(t0) >h and corresponding ∈ there is a neighborhood U of t0 so that f0(t) > h for all t U. It is immediate then that sup f(t) : f X > h if t U and this∈ verifies Part 4. { ∈ } ∈ Part 5 follows from Part 3 if X is a finite set. In general a sum of nonnegative functions is the supremum of all finite sums and so the general case follows from the specific by Part 4. Proposition 1.5. A nonnegative function f belongs to M + if and only if f(t) = sup g(t): g C+(T ),g f . { ∈ 0 ≤ } + + Proof. Certainly the sup g C0 (T ) : g f is in M by the property (4) noted above that{M +∈ is closed under≤ taking} supremums. Conversely, suppose that f M +. We wish to show that, for any ∈ + t0 T , f(t0) = sup g(t0) : g C (T ),g f . If f(t0) = 0 then ∈ { ∈ 0 ≤ } 4 1. INTEGRATION ON LOCALLY COMPACT SPACE this is certainly so and therefore we can suppose that f(t0) > 0. If 0 <h<f(t0) then there is a neighborhood U of t0 so that f(t) >h if t U. We may assume U is compact. By Urysohn’s Lemma ?? there ∈ + is g C0 (T ) so that g(t0) = h and g(t)=0if t / U. (We note U is normal,∈ as a compact Hausdorff space, so that Urysohn∈ ’s Lemma applies.) Since h<f(t0) is arbitrary we have f(t0) = sup g(t0) : g M +,g f . { ∈ ≤ } We now extend the integral I as the upper integral I defined on M + f by ∋ I(f) = sup I(g) g≤f + where the supremum is over the set Yf = g C0 (T ): g f . In the + { ∈ ≤ } special case that f C0 (T ) then I(f)= I(f) because f Yf . Thus I is an extension of I∈. It is further clear that, if f,g M +∈then ∈ f g implies I(f) I(g) ≤ ≤ and I(cf)= cI(f) if c> 0 Proposition 1.6. Suppose that X M + is directed upward (that is, for every f,g X, there is h X⊆so that h f and h g). ∈ ∈ + ≥ ≥ Then supg∈X g (the pointwise supremum) is in M and I(sup g) = sup I(g) g∈X g∈X Proof. Let f0 = supg∈X g and we have already seen earlier that + f0 M . It is clear that I(f0) = I(sup g) sup I(g) simply ∈ g∈X ≥ g∈X because f0 g if g X. It remains to prove the converse. Suppose ≥+ ∈ that h C (T ) and h f0. It suffices to show that sup I(g) ∈ 0 ≤ g∈X ≥ I(h). Choose ǫ > 0 and for each y in the support of h chose gy X so that g (y) > h(y) ǫ/2.