Integration on Locally Compact Space

CHAPTER 1

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 hf(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 continuous 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 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 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. This is possible because sup g(y) =∈ y − g f0(y) h(y). There is a neighborhood Uy of y so that gy(x) >h(y) ǫ/2 >≥ h(x) ǫ if x U because g is lower semi-continuous and− − ∈ y y h is continuous. Finitely many neighborhoods Uy1 ,...,Uyk cover the support of h. There is g X so that g max gyj :1 j k because X is directed upwards.∈ Consequently,≥ for any{ x in the≤ support≤ } of h, x Uyj for some j and therefore g(x) gyj (x) > h(x) ǫ. Let us ∈ + ≥ − suppose that u C0 (T ) is chosen so that u 1 on the support of h and 0 u 1.∈ Then ≡ ≤ ≤ I(g) I(ug) I(u(h ǫ)) = I(uh) ǫI(u)= I(h) ǫI(u) ≥ ≥ − − − It follows that sup I(g) I(h) and this completes the proof. g ≥ 1.THEINTEGRAL 5

Corollary 1.7. If X M + then I( f)= I(f) ⊆ fX∈X fX∈X

Proof. Consider the case that X = f1,f2 , that is X consists of + { } two elements. Then fi = sup g C0 (t) : g fi , for i = 1, 2 and { ∈ + ≤ } so f1 + f2 = sup g1 + g2 : g1,g2 C0 (T ),g1 f1; g2 f2 . By the preceding result { ∈ ≤ ≤ }

I(f1 + f2) = sup I(g1 + g2) = sup(I(g1)+ I(g2))

= sup I(g1)+ sup I(g2)= I(f1)+ I(f2) g1≤f1 g2≤f2 The result is true therefore if X consists of two elements or finitely many elements. In general I( f∈X f) is the supremum of I( f∈X′ f) where X′ X is a finite and soP the general case follows fromP the finite case by way⊆ of the preceding result. We further define the upper integral I∗(f) for any nonegative, extended valued function f : T [0, ] by → ∞ I∗(f) = inf I(g): g M +; f g = inf I(g): g B { ∈ ≤ } { ∈ f } + where Bf = g M : g f . Some of the{ ∈ properties≥ of }I∗ are: (a) I∗(f)= I(f) if f M +. ∗ ∗ ∈ (b) I (f1) I (f2), if 0 f1 f2. (c) I∗(cf)=≤ cI∗(f) if c>≤ 0. ≤ ∗ ∗ (d) supn I (fn)= I (supn fn) if 0 f1 f2 ...fn .... (e) I∗( f ) I∗(f ) if f ≤: T ≤ [0≤, ], n≤ N is any n n ≤ n n n → ∞ ∈ sequenceP of functions.P Parts (a), (b), and (c) are easily verified; we consider Part (d). If we ∗ ∗ ∗ set f0 = supn fn then I (fn) I (f0) for every n and so supn I (fn) ∗ ∗ ≤ ∗ ≤ I (f0) = I (supn fn). It therefore remains to show that supn I (fn) ∗ ∗ ≥ I (f0). We may suppose that sup I (fn) < because otherwise the inequality is obvious. Suppose that ǫ> 0 is given∞ and that for each n + ∗ n ∈ N, gn M , gn fn and I (fn) > I(gn) ǫ/2 which is possible since ∗ ∈ ≥ − I (fn) < . Define hn = max1≤i≤n gi so that hn forms an increasing ∞ + sequence in M . Consider h +1 + min h ,g +1 = max h ,g +1 + n { n n } { n n } min hn,gn+1 = hn +gn+1. Consequently I(hn+1)+I(min hn,gn+1 )= { } ∗ { } I(hn)+I(gn+1) and so I(hn+1)+I (fn) I(hn)+I(gn+1). Re-arranging, we have ≤ ∗ ∗ ∗ n+1 I(h +1) I(h ) I(g +1) I (f ) I (f +1) I (f )+ ǫ/2 n − n ≤ n − n ≤ n − n Adding over n, 1 n m gives ≤ ≤ ∗ ∗ I(h +1) I(h1) I (f +1) I (f1)+ ǫ/2. m − ≤ m − 6 1. INTEGRATION ON LOCALLY COMPACT SPACE

Taking the supremum and using the property that I commutes with the supremum, we have ∗ I(sup hm) sup I (fm)+ ǫ m ≤ m ∗ because I(h1) = I(g1) < I (f1) + ǫ/2, Of course f0 supm hm = ∗ ∗ ≤ supm gm because fm gm and so I (f0) supm I (fm)+ ǫ and, since ǫ> 0 is arbitrary, this≤ establishes Part (d).≤ ∗ We justify Part (e), beginning with the special case: I (f1 + f2) ∗ ∗ + ≤ I (f1)+ I (f2) if f1, f2 are nonegative functions. If g1,g2 M and ∈ f1 g1 and f2 g2 then ≤ ≤ ∗ I (f1 + f2) I(g1 + g2)= I(g1)+ I(g2) ≤ ∗ and if we take the infimum over g1 and g2 then we get I (f1 + f2) ∗ ∗ ≤ I (f1)+ I (f2). This implies Part (e) in the special case of finite sums and the general case follows by taking the supremum of the partial finite sums and applying Part (d) above. Definition A function f : T C is negligible if I∗( f )=0 (Alternatively a negligible function→ is called a zero function| |.) An increasing sequence of nonnegative, negligible functions has a negligible limit by Part (d) above. The sum of a countable collection of nonnegative, negligible functions is negligible by Part (e) above. We now consider subsets of T . The characteristic function of a subset A T is the function ⊆ 1 if t A, χA(t)= ∈ 0 if t / A. ∈ ∗ Definition: The exterior measure of a set A T is I (χA) ∗ ∗ ∗ ⊆ and is denoted µ (A): µ (A)= I (χA). A function f is negligible if and only if A = t : f(t) = 0 has exterior measure zero. For suppose that µ∗(A) = 0.{ Then 6 } ∗ ∗ ∗ ∗ I ( f )= I ( f χA) = sup I (min f ,n χA) sup nI (χA)=0 | | | | n {| | } ≤ n because f = sup min f ,n . Conversely suppose that I∗( f ) = | | n {| | } | | 0. For each n N define An = t : f(t) 1/n . Then 0 = ∗ ∗ ∈ 1 ∗ { | ∗| ≥ } ∗ I ( f ) I ( f χAn ) n I (χAn ) Therefore µ (An) = 0 and µ (A) = | | ∗ ≥ | | ≥ supn µ (An)=0. Definition A set A T is negligible (or a zero set) if the ⊆ characteristic function χA is negligible. If An is a sequence of sets then ∗ ∗ ∗ ∗ ∗ µ ( A )= I (χ∪ ) I ( χ ) I (χ )= µ (A ) ∪n n nAn ≤ An ≤ An n Xn Xn Xn 1.THEINTEGRAL 7

If we further assume that the An are pairwise disjoint open sets then µ∗( A )= µ∗(A ) ∪n n n Xn because the characteristic function of an open set is lower semi-continuous and so so µ∗( A )= I( χ ) and, of course I is additive. ∪n n n An We can summarize someP of these properties. Let An be a sequence of sets in T . (i) µ∗( A ) µ∗(A ). ∪n n ≤ n Xn (ii) µ∗( A )= µ∗(A ) if the A are open and pairwise dis- ∪n n n n Xn joint. (iii) nAn is negligible if each An is. ∪ ∗ ∗ (iv) If A1 A2 ... An ... then µ ( nAn) = lim µ (An) v If I∗(f⊆) < ⊆, where⊆ f ⊆ 0 then f(t)=∪ is negligible. ∞ ≥ { ∞} Properties (i) through (iv) follow from the earlier noted properties of I∗. Property (v) can be verified as follows: we set A = f(t)= , ∗ ∗ ∗ { ∞} and observe > I (f) I (nχA) = nµ (A) for every n > 0 and so µ∗(A) = 0. Further∞ we have≥ two more properties. (vi) The exterior measure of a compact set is finite. (vii) If a set A has finite exterior measure then, for each ǫ> 0 there is an open set U A so that µ∗(U) µ∗(A)+ ǫ. ⊇ ≤ For, if Q is compact, then there is a function f C0(T ) so that f is identically one on Q and so µ∗(Q) I∗(f) = I(f)∈< which proves (vi). The existence of f is guaranteed≤ by Urysohn’s Lemma∞ ??. Now suppose that a set A has exterior measure µ∗(A) < and + ∗ ∗ ∞ δ > 0. There is g M so that g χA and I (g) < I (χA)+ δ = µ∗(A)+ δ. Define U∈= g > 1 δ so≥ that U is open because g is lower semi-continuous and U{ A and− (1} δ)µ∗(U) I∗(g) <µ∗(A)+ δ and since δ > 0 is arbitrary,⊇ this verifies− (vii). ≤ Let T be a locally compact space and define E to be the set of all functions f defined and complex valued on a set D T such that T D is negligible. The set D depends on f in general. Then⊆ E includes,\ for example, extended real valued functions provided those functions are finite valued (real valued) on all but a negligible set. The set E is a linear space over C under the usual operation of scalar multiplication and addition of functions defined as follows: If f,g E are defined and complex valued on D and D where T D and∈T D are negli- f g \ f \ g gible then f + g is defined and complex valued on Df Dg and the complement (T D ) (T D ) is negligible by the preceding∩ result. (It \ f ∪ \ g 8 1. INTEGRATION ON LOCALLY COMPACT SPACE is not necessary to concern ourselves with how f + g might be defined on the negligible set.) Thus E is a linear space. On E we define a relation f g if f = g on all but a negligible set. Equivalently f g if f g is negligible.∼ It is clear that this defines an equivalence relation.∼ |(In− other| words it is a reflexive, symmetric and transitive relation.) This equivalence relation is compatible with the arithmetic operations of functions or, in other words, f1 g1 and f2 g2 and c C im- ∼ ∼ ∈ plies f1 + f2 g1 + g2, cf1 cg1 and f1f2 g1g2. If[f] denotes the equivalence class∼ that contains∼ f E then∼ we can define c[f]=[cf] ∈ and [f1]+[f2]=[f1 + f2] and this is well defined, that is it does not depend on the particular choice of representative in the equivalence class. Observe that each equivalence class contains a function that is defined and complex valued on all of T . For example, [f] contains the function f˜ which is defined to be equal to f wherever f is defined and complex valued and f˜ is zero elsewhere. The linear space E/ consisting of all equivalence classes in E. ∼ An example of the convenience of working in E/ is that if f1 and f2 are extended real valued functions in E (so that they∼ are finite valued on all but a negligible set) can be added without concern about the indeterminant +( ): we can ignore what happens on a negligible set. ∞ −∞ If f g and f,g 0 then I∗(f) = I∗(g) because I∗(f) I∗(g + f g ) ∼ I∗(g)+ I∗(≥f g )= I∗(g). Similarly I∗(g) I∗(f≤) and so we| − have| ≤I∗(f)= I∗(g).| Therefore− | if f 0 almost everywhere≤ then we may define I∗([f]) = I∗(f) and this integral≥ is well defined. Definition Any property is said to hold almost everywhere if the property does hold except possibly on a negligible set. For example, an extended valued function f is finite almost everywhere if t : f(t) = is negligible. A second example is f g if f = g almost{ | everywhere.| ∞} ∼ For f 0 and p, 1 p< , define ≥ ≤ ∞ N (f)= I∗( f p)1/p p | | Then (1.1) N ( f ) N (f ) p n ≤ p n Xn∈I Xn∈I if I is a countable index set. Indeed if p = 1 then this follows from ∗ Property (e) of I . If p = 2 and then N2(f1 +f2) N2(f1)+N2(f2) follows from the Cauchy-Schwarz-Bunyakovsky inequality.≤ The inequality is therefore clear when I is a finite index set and so the general case follows by Property (d) of I∗. For general p, 1 p< the inequality ≤ ∞ 1.THEINTEGRAL 9 can be verified by establishing the Minkowski inequality in the present context as in Theorem ??. (In Theorem ?? the integral is countably additive unlike I∗ and the functions here need not be measurable.) The cases p = 1, 2 are the only cases that we shall consider subsequently and so the case of general p is not pursued here. Let us remark that Np(f) = 0 if and only if f = 0 almost ev- p erywhere. Indeed Np(f)=0iff f is negligible iff f is zero almost everywhere. We define | | p = f E : N (f) < F { ∈ p ∞} (Recall E is the space of functions defined and complex valued almost everywhere on the locally compact space T .) Theorem 1.8. For 1 p< ≤ ∞ (1) p is a vector space over C; F p (2) Np is a semi-norm on ; (3) p is complete with respectF to the semi-norm N . F p Again the cases p = 1, 2 are the only cases that we shall consider subsequently. Certainly Np is a semi-norm by 1.1. (Of course one must observe further that Np(cf) = c Np(f) for any c C.) Consequently p is a linear space. | | ∈ F Completeness can be verified as in Theorem ??. p p p We define (T ) or to be the closure of C0(T ) in . The functions f L p(T ) areLp-integrable functions (or simply integrableF when p = 1).∈ WeL include in p(T ) all functions g which are defined almost everywhere on T andL which agree almost everywhere with a function defined on all of T and in p(T ). L We recall that I(f) I( f )= N1(f) if f C0(T ). This says that the integral I is a continuous| |≤ | linear| functional∈ defined on the subspace 1 1 C0(T ) of . We extend I from C0(T ) to by continuity so that, if 1 L L f then there is a sequence fn C0(T ) so that limn N1(f fn)=0 and∈L we define I(f) = lim I(f ). Clearly∈ 1 is a complex linear− space n n L with norm N1 and I is a continuous linear functional on it. A function in 1 is said to be summable. L 1 Of course the semi-norm N1 is also continuous on and if f ∗ ∗ L ∈ C0(T ) and f 0 then N1(f)= I ( f )= I (f). Consequently, ≥ | | (1) I∗(f)= I(f) if f 1 and f 0. Also 1 ∈L1 ≥ (2) If f then f and I(f) I( f )= N1(f). ∈L | |∈L ≤ | | For suppose that f C0(T ) is a sequence that converges to f. n ∈ Then fn f fn f so that f is also in the closure of C0(T ) || |−| || ≤ | − | | | 1 and I( f ) = lim I( f ) so that f and I(f) I( f )= N1(f). | | n | n| | |∈L ≤ | | 10 1. INTEGRATION ON LOCALLY COMPACT SPACE

(3) If f,g 1 then min f,g =(f + g f g ) /2 and max f,g = (f + g∈L+ f g ) /2 are{ also} in 1. −| − | { } (4) If f is negligible| − | then f 1 andL I(f)=0. ∈L for we simply observe that f is the limit in the N1 norm of the identically 0 sequence in C0(T ). (5) Monotone Convergence: If fn is a nondecreasing (resp. nonincreasing) sequence of functions in 1 and if the real sequence I(f ) converges then f = lim fL belongs to 1 and n n n L I(f) = limn I(fn).

Certainly the pointwise limit f = limn fn exists, because monotone sequences converge but of course f may be infinite valued. We consider the case fn is nondecreasing. The sequence fn f1 is nonnegative and ∗ ∗ − ∗ increasing and so sup I (f f1) = I (f f1), by Property 4 of I . n n − − This shows that f f1 is the N1 limit of the sequence fn f1 and so 1 − − f f1 and also I(f f1) = lim I(f f1) (by the continuity of − ∈L − n n − I) and, by linearity, I(f) = limn I(fn). Further we have 1 (6) If fn and fn 0 and n I(fn) < then n fn exists ∈ L ≥ ∞ 1 almost everywhere and definesP a function f Pand I(f) = ∈ L n I(fn). + (7) PIf g M (that is g is lower semi-continuous and nonnega- ∈ 1 tive) then g if and only if I(g) < and in this case ∈ L ∞ I(g)= I(g). Property (6) follows by applying (5) to the partial sums. As for 1 ∗ (7), certainly g implies I(g) < because I(g) = I (g) = N1(g) ∈ L ∞ by Property (1) and I(g) = I(g). It remains to check that I(g) < 1 ∞ implies g . We recall that I(g) = sup I(f) where the supremum ∈L f≤g is over all f C0(T ) so that 0 f g. Consequently we can choose a ∈ ≤ ≤ sequence fn C0(T ) so that 0 fn g and I(fn) > I(g) 1/n. Since +∈ ≤ ≤ − g fn M we have 0 = limn I(g fn) = limn N1(g fn) which says − ∈ − 1 − that g is in the closure of C0(T ), that is, in . L (8) If g is upper semi-continuous, finite valued and g 0 then g 1 if and only if I∗(g) < and in this case I∗(g≥)= I(g). ∈L ∞ Clearly g 1 implies I∗(g)= I(g) < . Conversely, if I∗(g) < , ∈L ∞ ∞ then there is a sequence of lower semi-continuous functions gn g so ∗ ∗ ∗ ≥ that limn I (gn) = I (g). In particular I (gn) < at least for large 1 ∞ + 1 n so that gn (by (7)) and similarly gn g M is in and, consequently ∈g = L g (g g) is in 1. (Actually− ∈ g = g (Lg g) n − n − L n − n − only when gn is finite valued but that is almost everywhere.) (9) If f 0 and f 1 and ǫ > 0 then there is a non-negative upper≥ semi-continuous∈ L function h with compact support and 1.THEINTEGRAL 11

a lower semi-continuous function g so that h f g almost everywhere and ≤ ≤ I(g h) <ǫ − If we further suppose that there is a finite-valued, lower semi- continuous function G f everywhere then h and g can be chosen so that h f ≥g everywhere (not just almost everywhere) and I(g ≤h) <ǫ≤ − There is a function k C0(T ) so that I( f k ) < ǫ/4 because f 1. By the definition∈ of I∗ there is a function| − |g ˜ M + so that f ∈k L g˜ almost everywhere and I∗(˜g) < ǫ/2. (We may∈ supposeg ˜ is finite-valued| − |≤ in the case that there is a lower semi-continuous function G f for otherwise we can replaceg ˜ by min g,G˜ + k because f ≥k G+ k and G+ k is finite-valued.) Consequently{ | max|} 0, k |g˜ − f|≤ k +˜|g |and so if we| | set h = max 0, k g˜ and g = k +˜g {then−h is}≤ upper≤ semi-continuous and compactly{ supported− } whereas g is lower semi-continuous and h f g almost everywhere and g h 2˜g so that I(g h) <ǫ. ≤ ≤ − ≤ Remark− It is not true as suggested in Naimark’s book (Normed Algebras) that if f 0 is finite-valued (or f k ) then there is a finite- valued lower semi-continuous≥ functiong ˜ so| that− | f < g˜. For example if f(x)=1/√x and f(0) = 0 theng ˜(x) 1/√x andg ˜(0) = . ≥ ∞ (10) If f 0 and f 1 and ǫ > 0 then there is a compact set ≥ ∗∈ L Q T so that I (f(1 χQ)) <ǫ. (11) If ⊆f 0 and f 1−then there is an increasing sequence ≥ ∈ L hn of upper semi-continuous nonnegative functions of compact support and a decreasing sequence gn of lower semi-continuous functions so that h f g almost everywhere and n ≤ ≤ n lim I(hn)= I(lim hn)= I(f)= I(lim gn) = lim I(gn) n n n To establish Property (10) we let Q be the compact support of the ∗ function h of Property (9) . Then ǫ>I(f h) I (χQc (f h)) = ∗ − ≥ − I (fχQc ). For Property (11), we apply Property (9) to choose hñ andg ñ corresponding to ǫ =1/n, for each n N. Then we define h = max h˜ : ∈ n { j 1 j n and gn = min g˜j : 1 j n . The convergence of the integrals≤ ≤ follows} by Property{ (5): monotone≤ ≤ } convergence.

Index

characteristic function , 6 exterior measure , 6 negligible , 6 zero function , 6 integral, 1