Appendix A Compact and Totally Bounded Metric Spaces
Definition A.0.1 (Diameter of a set) The diameter of a subset E in a metric space (X, d) is the quantity
diam(E) = sup d(x, y). x,y∈E
♦
Definition A.0.2 (Bounded and totally bounded sets)A subset E in a metric space (X, d) is said to be 1. bounded, if there exists a number M such that E ⊆ B(x, M) for every x ∈ E, where B(x, M) is the ball centred at x of radius M. Equivalently, d(x1, x2) ≤ M for every x1, x2 ∈ E and some M ∈ R; > { ,..., } 2. totally bounded, if for any 0 there exists a finite subset x1 xn of X ( , ) = ,..., = such that the (open or closed) balls B xi , i 1 n cover E, i.e. E n ( , ) i=1 B xi . ♦
Clearly a set is bounded if its diameter is finite. We will see later that a bounded set may not be totally bounded. Now we prove the converse is always true.
Proposition A.0.1 (Total boundedness implies boundedness) In a metric space (X, d) any totally bounded set E is bounded.
Proof Total boundedness means we may choose = 1 and find A ={x1,...,xn} in X such that for every x ∈ E
© The Editor(s) (if applicable) and The Author(s), under exclusive 429 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 430 Appendix A: Compact and Totally Bounded Metric Spaces
inf d(xi , x) ≤ 1. 1≤i≤n
Set M = max1≤i≤n d(x1, xi ). We know there is an index i ∈{1,...,n} such that d(xi , x) ≤ 1, so the triangle inequality implies
d(x1, x) ≤ d(x1, xi ) + d(xi , x) ≤ M + 1.
Hence E ⊆ B(x1, M + 1), and E is bounded.
Example A.0.1 (Bounded versus totally bounded sets) The notions of boundedness and total boundedness differ, because the former is weaker. The discrete distance of distinct points is always 1, and equals 0 for coinciding points. • The space X ={a, b, c} with discrete metric is surely bounded. It is totally bounded, too, since {a}, {b}, {c} have zero diameter (less than any > 0), there is a finite number of them, and their union is X. • The space [0, 1] equipped with the discrete distance, rather than the usual Euclidean metric, is bounded: its diameter is 1, the distance of any pair of distinct points. But it is not totally bounded. The subsets of diameter less than 1 are the singletons {x}, of which there is no finite number covering the whole space. • N with the discrete metric is bounded (distinct natural numbers are all 1 apart), but not totally bounded. We cannot cover N by finitely many balls of radius < 1 (a ball consists of a single point, its centre, but N is infinite). A Cauchy sequence of natural numbers is eventually constant, hence it converges. N is therefore a complete metric space.
Proposition A.0.2 (Separability of totally bounded metric spaces) Totally bounded metric spaces (X, d) are separable.
Proof As X is totally bounded, for every n ∈ N there exists An ={x1,...,xn}⊆X, finite, such that n 1 X = B x , . i n i=1
∞ The union D = = An is countable (as countable union of finite sets). n¯ 1 We claim that D = X.Letx ∈ X be an arbitrary point. For any n ∈ N, there ∈ ∈ , / { } ⊆ exists some xin An such that x B xin 1 n . This gives a sequence xin n≥1 D Appendix A: Compact and Totally Bounded Metric Spaces 431
( , )< / = ∈ with d x xin 1 n. Therefore lim xi x. This shows that every x X lies in the closure of D and so D¯ = X.
Intuitively a separable space is one where any element is close to one from some (dense) countable subset. In a compact space, instead, any element is close to another taken from a finite number of subsets, so compact spaces can be considered kind of “pseudo-finite”. In analogy to the notion of compactness in R we have the following definition for generic metric spaces.
Definition A.0.3 (Compactness) A subset E in a metric space (X, d) is said to be compact if any open cover of E admits a finite subcover (so-called Heine–Pincherle– Borel property). ♦
The next results generalises the Bolzano–Weierstraß theorem to compact metric spaces.
Theorem A.0.1 (Bolzano–Weierstraß theorem for metric spaces) If (X, d) is a com- pact metric space, any infinite subset E ⊆ X has a limit point. Proof Let (X, d) be compact and E ⊆ X infinite. Assume, by contradiction, E has no limit points. Then for every x ∈ X there is an open ball B(x, rx ) such that B(x, rx ) ∩ E ⊆{x}. The family {B(x, rx )}x∈X is an open cover of X, so there exists ( , ) = ,..., a finite subcover B xi rxi , i 1 n. But then n = ∩ = ∩ ( , ) ⊆{ ,..., }, E E X E B xi rxi x1 xn i=1 i.e. E is finite. The contradiction proves the claim.
Definition A.0.4 (Sequentially compact metric spaces) A metric space (X, d) is sequentially compact if every sequence in X admits a convergent subsequence. ♦
We will see straightaway that sequential compactness implies total boundedness.
Proposition A.0.3 (Sequential compactness versus total boundedness) A sequen- tially compact metric space (X, d) is totally bounded. 432 Appendix A: Compact and Totally Bounded Metric Spaces
Proof Suppose (X, d) is not totally bounded. Then there is an 0 such that X cannot be covered by balls centred at the points of any finite set A0 ⊆ X. Taking x1 ∈ X, then, we can find x2 ∈ X with d(x1, x2)>0, for otherwise X = B(x1, 0) and A0 ={x1}. Next, take x1, x2 ∈ X, and there is an x3 ∈ X with d(x3, x1)>0 and d(x3, x2)> 0, because otherwise X = B(x1, 0) ∪ B(x2, 0) and A0 ={x1, x2}. Inductively, at step k + 1wehave{x1,...,xk }⊆X and there is an xk+1 ∈ X with d(xk+1, xi )>0 for every i ∈{1,...,k}, because if not, once again,
k X = B(xi , 0) i=1 and A0 ={x1,...,xk }. Overall we have a whole sequence {xk }k≥1 such that d(xk , x j )>0, k = j.It must diverge, hence it certainly cannot have a convergent subsequence. But then X would not be sequentially compact, contradicting the hypothesis.
Definition A.0.5 (Countably compact metric spaces) A metric space (X, d) is countably compact if any countable open cover of X has a finite subcover. ♦
Proposition A.0.4 (Sequentially compact metric spaces and countable compact- ness) A sequentially compact metric space (X, d) is countably compact.
Proof Let {Gn}n≥1 be a countable open cover of X ∞ X = Gn. n=1
Suppose X not countably compact, so {Gn}n≥1 does not have a finite subcover, meaning for every m ≥ 1
m Gn ⊆ X. n=1 ∈ ∈/ m { } The proof is by induction. Take xm1 X such that xm1 n=1 Gn, and since Gn n≥1 ∈ ∈ ∈/ covers X, there is an m1 such that xm1 Gm1 . Choose xm2 X such that xm2 m1 ∈ ∈ n=1 Gn. Then there is an m2 such that xm2 Gm2 . Then pick xm3 X such that ∈/ m2 ∈ xm3 n=1 Gn,soxm3 Gm3 for some index m3. At step k Appendix A: Compact and Totally Bounded Metric Spaces 433
mk−1 ∈ ∈/ . xmk Gmk and xmk Gn n=1 { } We end up with a sequence xmk k≥1 in the sequentially compact space X, so there { } ∈ { } is a subsequence xkn n≥1 with some limit point x X.But Gn n≥1 is a cover, so x belongs in some Gn0 . Furthermore,
lim xk = x n→∞ n ∈ ≥ forces xkn Gn0 for all n n0. But because of how the sequence was constructed, − > ∈ ∈/ ⊆ mkn 1 there exists an mkn n0 such that xkn Gmkn , and xkn Gn0 n=1 Gn. Hence { } no subsequence of xmk k≥1 can converge to x. But this means X is not sequentially compact which is a contradiction. Therefore X is countably compact.
In metric spaces compactness can be characterised by means of sequences. Namely,
Theorem A.0.2 (Characterisation of compact metric spaces in terms of sequential compactness) A metric space (X, d) is compact if and only if it is sequentially compact.
Proof (⇒)Let(X, d) be compact, and let us prove it is sequentially compact. Given a compact subset E ⊆ X, suppose there is a sequence {xn}n≥1 none of whose subsequences converge in E. Any such must have infinite range, otherwise it would become eventually constant and hence converge in E. Nor can it accumulate at a point in E, otherwise this point would be the limit of a subsequence. Take x ∈ E. This is not a limit point for {xn}n≥1,sothereisanx > 0 such that the ball B(x, x ) contains at most one term of the sequence. ( , ) Clearly B x x x∈E is an open cover of E, and by compactness we extract a finite subcover such that
n ⊆ ( , ). E B xi xi i=1 { } ⊆ n ( , ) But xn n≥1 E, and since in i=1 B xi xi there are at most n distinct points, we deduce the sequence can only attain finitely many values. Hence it must have a convergent subsequence, contradicting the hypothesis. (⇐) Consider a set of indices I and a family of open sets G ={Gi }i∈I with X = Gi . i∈I 434 Appendix A: Compact and Totally Bounded Metric Spaces
As X is sequentially compact, it is totally bounded (Proposition A.0.3), and by Proposition A.0.2 also separable. Call D ={x1,...,xn,...}⊆X the countable dense subset. Using this we con- struct the countable collection of balls B = B(xn, q) : xn ∈ D, q ∈ Q .
Take the subcollection BG = B(xn, q) ∈ B :∃Gi ∈ G such that B(xn, q) ⊆ Gi .
Clearly the latter is at most countable. At this point it is enough to show BG is an open cover of X. If so, in fact, by Proposition A.0.4 and Definition A.0.5 we can extract from it a finite subcover B(x1, q1),...,B(xk , qk ) . Each B(x j , q j ), j = 1,...,k, is contained in some Gi of the original cover G. Renaming these G j gives a finite subcollection {G j : 1 ≤ j ≤ k} of G such that
k X = G j , j=1 whence compactness follows. So take an x in X.AsG covers X, x belongs to some Gi . The latter is open, so for > 0 there is an open ball B(x, ) ⊆ Gi . But D is dense in X, so there exists xn ∈ D such that d(x, xn)</2, i.e. x ∈ B(xn, /2). Moreover, for every y ∈ B(xn, /2) we have d(y, xn)</2, so
d(y, x) ≤ d(y, xn) + d(x, xn)< from the triangle inequality. Hence
x ∈ B(xn, /2) ⊆ B(x, ) ⊆ Gi .
Pick q ∈ Q such that /2 < q < , so that
x ∈ B(xn, q) ⊆ B(x, ) ⊆ Gi .
As B(xn, q) has rational radius and centre xn ∈ D, it belongs to the family B.But B(xn, q) is a subset of Gi , so it also belong to BG. In conclusion, any x ∈ X belongs to a ball of BG, making the latter an open countable cover of X.
There is a second description of compactness in metric spaces in terms of complete- ness and total boundedness. Appendix A: Compact and Totally Bounded Metric Spaces 435
Theorem A.0.3 (Characterisation of compact metric spaces in terms of complete- ness and total boundedness) A metric space (X, d) is compact if and only if it is complete and totally bounded.
Proof If X is compact, it is sequentially compact by what we said previously. Then X is complete and by Proposition A.0.3 it is totally bounded as well. Sufficient implication: suppose X complete, totally bounded and not compact, by ( , ) contradiction. Hence it has an open cover B xα rα α∈I without finite subcovers. = { ,..., } Total boundedness guarantees that for 1 there exists a finite set x11 xn1 ( , ) = ,..., such that the (open or closed) balls B xi1 1 , i 1 n, cover X.Call x1 the ele- { ,..., } ( , ) ment in x11 xn1 with the property that no finite subcollection of B xα rα α∈I is a cover of B(x1, 1). Such an element must exist, because if the balls of radius 1 ,..., ( , ) centred at x11 xn1 possessed a finite subcover extracted from B xα rα α∈I then X would be compact. = / ,..., Similarly, choosing 1 2weletx2 be the element among x11/2 xn1/2 for which B(x2, 1/2) ∩ B(x1, 1) =∅and B(x2, 1/2) is not covered by any finite sub- family of the open cover of X.Thisx2 exists, too: if all balls of radius 1/2 centred ,..., ( , ) at x11/2 xn1/2 and intersecting B x1 1 had a finite subcover of the open cover, then B(x1, 1) would admit a finite subcover of the original family. = / m ,..., By induction, for 1 2 we call xm the element among x11/2m xn1/2m such m m−1 that B(xm , 1/2 ) ∩ B(xm−1, 1/2 ) =∅and with the property that no finite sub- m family of the open cover of X is a cover of B(xm , 1/2 ). m m−1 Take x ∈ B(xm , 1/2 ) ∩ B(xm−1, 1/2 ), so the triangle inequality forces 1 1 1 d(x − , x ) ≤ d(x − , x) + d(x , x) ≤ + ≤ , m 1 m m 1 m 2m−1 2m 2m−2 and for k < m
d(xk , xm ) ≤ d(xk , xk+1) + d(xk+1, xk+2) + ...+ d(xm−1, xm ) ≤ 1 + 1 + 1 + ...+ 1 ≤ 1 . 2k−1 2k 2k+1 2m−2 2k−2
Hence {xi }i≥1 is a Cauchy sequence in the complete space X, so it converges to a point x ∈ X. ( , ) ( , ) Call B xα0 rα0 the ball of the open cover B xα rα α∈I that contains x. There > ( , ) ⊆ ( , ) exists an 0 such that B x B xα0 rα0 . By construction, moreover, there is m an m such that 1/2 < /2 and d(xm , x)</2, so finally
( , / m ) ⊆ ( , ) ⊆ ( , ). B xm 1 2 B x B xα0 rα0
m But then one ball from the open cover is enough to cover B(xm , 1/2 ), contradicting the assumption. 436 Appendix A: Compact and Totally Bounded Metric Spaces
Theorems A.0.2 and A.0.3 imply that a metric space is sequentially compact iff it is complete and totally bounded.
Proposition A.0.5 (Compactness in complete metric spaces) In a complete metric space (X, d) a subset F is compact if and only if it is closed and totally bounded.
Proof The ⇒ part is self-evident: F is compact (also sequentially) and it contains all of its limit points. Conversely, take a sequence {xn}n≥1 in F. By assumption F is totally bounded, so for every = 1/m, m ≥ 1, F can be covered by finitely many balls B(xn, 1/m) = {y ∈ X : d(xn, y) ≤ 1/m}, n = 1,...,k. = { ,..., } Choosing 1, let x11 xk1 denote the subset of X such that the closed ( , ) balls B xi1 1 of radius 1 finitely cover F. Among these there is at least one that { } ⊆ = ( , ) contains infinitely many terms of xn n≥1 F, call it B1 B xn1 1 . Then define N1 ={n : xn ∈ B1}, an infinite set. ∈ = / { ,..., } Pick n1 N1 and 1 2. Let x12 xk2 be the finite subset of X for which ( , / ) = ( , / ) the closed balls B xi2 1 2 cover F. One of these, say B2 B xn2 1 2 , contains infinitely many terms of {xn}n≥1 ⊆ F. Define N2 ={n > n1 : xn ∈ B1 ∩ B2} and so forth. { } { } ≥ Eventually we obtain a subsequence xni i≥1 of xn n≥1 such that, for every n j = ( , / ) { } ni ,thetermsxn j belong to Bi B xni 1 i , and therefore xni i≥1 is Cauchy in Bi ⊆ F. By completeness {xn}n≥1 converges in X, and as a matter of fact in F, since F is closed. Thus we proved {xn}n≥1 ⊆ F has a convergent subsequence, and F is compact.
The Heine–Pincherle–Borel theorem descends in a straightforward manner from the the above result.
Theorem A.0.4 (Heine–Pincherle–Borel theorem for complete metric spaces) Let (X, d) be a complete metric space and F ⊆ X a closed and totally bounded subset. Any cover of open balls {B(xα, rα)}α∈I of F admits a finite subcover.
At last, here is the version of Cantor’s intersection theorem for complete metric spaces. Appendix A: Compact and Totally Bounded Metric Spaces 437
Theorem A.0.5 (Cantor’s intersection theorem in complete metric spaces) In a com- plete metric space (X, d) the intersection of a decreasing sequence of non-empty, closed and totally bounded subsets is non-empty.
Theorem A.0.6 ( Cardinality of perfect sets) In a complete metric space (X, d) any non-empty, perfect and totally bounded set P ⊆ X has the cardinality of the continuum.
Proof Let P be a perfect, non-empty and totally bounded set in X. As any point in P is a limit point of P, the set must be infinite. Suppose P is countable. Take one point x1 and B(x1, r1) = x ∈ X : d(x, x1) As x1 is a limit point of P, B(x1, r1) contains infinitely many points of P different ¯ ¯ from x1,soK1 = B(x1, r1) ∩ P =∅, where B(x1, r1) = x ∈ X : d(x, x1) ≤ r1 . Among those pick x2 = x1 and let r2 > 0 be such that r2 < min d(x1, x2), r1 − d(x1, x2) . Call B(x2, r2) = x ∈ X : d(x, x2) ¯ ¯ ¯ so B(x2, r2) ⊆ B(x1, r1). Since d(x1, x2)>r2, then x1 ∈/ B(x2, r2). Furthermore, x2 is a limit point of P,soB(x2, r2) contains infinitely many points of P other than x2, ¯ and K2 = B(x2, r2) ∩ P =∅. Among them pick x3 = x2, and so forth. ¯ Suppose, by induction, to have built B(xn, rn) such that B(xn, rn) ∩ P =∅con- tains infinitely many points of P different from xn. Choose xn+1 = xn and take rn+1 < min d(xn, xn+1), rn − d(xn, xn+1) . Then B(xn+1, rn+1) = x ∈ X : d(x, xn+1) The set in 3. satisfies the induction hypothesis, so we can proceed with the construc- tion. ∈/ Since xn Kn+1, at the end of the process none of the points of P belongs in ∞ ⊆ ∞ =∅ { } n=1 Kn. Moreover Kn P for every n,so n=1 Kn .But Kn is a decreasing sequence of non-empty compact sets (Proposition 2.1.4) and with empty intersection. This violates Theorem A.0.5, and the claim follows. Appendix B Urysohn’s Lemma and Tietze’s Theorem In a metric space (X, d) any two non-empty, closed and disjoint sets A, B ⊆ X can be separated by open disjoint sets U, V ⊆ X. This is called normality property. Proving that metric spaces are normal is rather easy. Take in fact a ∈ A, b∈ B and ra = ( 1/3) · d(a, B)>0, rb = (1/3) · d(b, A)>0, where d(a, B) = inf d(a, b) : b ∈ B and d(b, A) = inf d(b, a) : a ∈ A . Then U = B(a, ra) and V = B(b, rb) a∈A b∈B are open, and disjoint since if y ∈ B(a, ra) ∩ B(b, rb) (assuming ra ≥ rb without loss of generality), the triangle inequality would give d(a, b) ≤ d(a, y) + d(y, b) ≤ ra + rb ≤ 2ra, whilst by construction d(a, b) ≥ 3ra. We set out to prove a general fact concerning normal topological spaces—hence valid for metric spaces—called Urysohn lemma. The objective is to build a continuous function f on a normal space X mapping non-empty closed subsets A, B to 0, 1 respectively. To do this we need a very large family of open sets containing A that do not meet B, and then decide which value f takes of a given x ∈ X by looking at which open sets it belongs. Theorem B.0.1 (Urysohn’s lemma1) Let X be a normal space and A, B non-empty, closed, disjoint subsets. Call U the family of neighbourhoods of A that do not intersect 1Pavel Samuilovich Urysohn (1898–1924) was born in Odessa. He studied under D.F. Egorov and N.N. Lusin in Moscow, where he was awarded a Ph.D. in 1921. Urysohn was one of the most promising Soviet mathematicians of his generation at the time of his death, at the age of 25, in a tragic accident while swimming off the coast of Brittany. Urysohn’s lemma, albeit trivial in metric spaces, allows to generalise Tietze’s extension theorem to normal spaces. © The Editor(s) (if applicable) and The Author(s), under exclusive 439 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 440 Appendix B: Urysohn’s Lemma and Tietze’s Theorem B, and D the set of rationals of the form m/2n,m, n ∈ N and 0 ≤ m/2n ≤ 1. Then there exists a continuous map f : X →[0, 1] such that f (A) = 0, f (B) = 1, and 0 < f (x)<1 if x ∈ X \ (A ∪ B). Proof The set m m 1 1 3 1 3 5 7 D = : m, n ∈ N and 0 ≤ ≤ 1 = 0, 1, , , , , , , ,... , 2n 2n 2 4 4 8 8 8 8 consists of the dyadic numbers in [0, 1]. For each q ∈ D take a neighbourhood Uq ⊆ X. Initially, choose U1 = X.AsX is normal, there are open disjoint neighbourhoods ¯ ¯ UA, UB such that UA ∩ UB =∅,soUA ∩ B =∅and UA ⊆ (X \ B) ⊆ U1 = X. Put U0 = UA, and by normality we have ¯ ¯ U0 ⊆ U1/2 ⊆ U1/2 ⊆ (X \ B) ⊆ U1 = X. Iterating in this way we can find U1/4 such that ¯ ¯ U0 ⊆ U1/4 ⊆ U1/4 ⊆ U1/2. Analogously, we can find U3/4 satisfying a similar relationship with U1/2 and (X \ B), and so on. By induction we end up with 1. A ⊆ Uq for every q ∈ D; 2. B ⊆ U1 and B ∩ Uq =∅for every q < 1; ¯ 3. Up ⊆ Uq for every p < q in D. Now define f : X →[0, 1] by f (x) = inf t ∈ D : x ∈ Ut . This map equals 0 on A (condition 1. above) and 1 on B (condition 2.). We need to show f is continuous, i.e. that the pre-image of any open subset in [0, 1] is open in X. Suppose V = (a, b) intersects [0, 1].Givenx ∈ f −1(V ), clearly f (x) ∈ (a, b), so there exist rational numbers p, q such that a < p < f (x) ¯ Pick y ∈ U, so by construction y ∈ Uq ⊆ Uq and by definition f (y) ≤ q < b. ¯ On the other hand by construction y ∈/ Up ⊇ Up, and by definition f (y) ≥ p > a. Hence f (y) ∈[p, q]⊆(a, b). Clearly f : X →[0, 1] is 0 on A and 1 on B, and it is continuous on X,sofor x ∈ X \ (A ∪ B) it must assume all values in (0, 1). Example B.0.1 (Example of Urysohn function on a metric space)TakeA, B non- empty, closed and disjoint in a metric space (X, d), and let us construct f : X → [0, 1] such that f (x) = 0 for every x ∈ A, f (x) = 1 for every x ∈ B and 0 < f (x)< 1 for every x ∈ X \ (A ∪ B). Define continuous maps (see Lemma 5.1.2) d(x, A) = inf d(x, a) : a ∈ A and d(x, B) = inf d(x, b) : b ∈ B . Clearly = 0ifx ∈ A = 0ifx ∈ B d(x, A) = and d(x, B) = > 0ifx ∈ Ac > 0ifx ∈ Bc. Since d(x, A) + d(x, B) is always positive and continuous, we introduce the conti- nuous map d(x, A) f (x) = . d(x, A) + d(x, B) In practice ⎧ ⎨ = 0ifx ∈ A ( ) = ∈ , f x ⎩ 1ifx B ∈ (0, 1) if x ∈ X \ (A ∪ B) furnishing the required function. Theorem B.0.2 (Tietze’s extension theorem2) Let X be a normal topological space and E a closed subset. Then 2Heinrich Tietze (1880–1964) was born in Schleinz, today’s Austria. He studied in Vienna, where he obtained his Habilitation in 1908. From 1910 until 1919 he was professor in Brno, during which time he proved the extension theorem for metric spaces. He spent the rest of his career teaching at universities in Erlangen and Munich. 442 Appendix B: Urysohn’s Lemma and Tietze’s Theorem 1. a continuous map f : E →[c, d] extends to a continuous map g : X →[c, d]; 2. a continuous map f : E → R extends to a continuous map g : X → R. ‘Extends’ means f (x) = g(x) for every x ∈ E. Proof We start by 1. Replacing f with ( f − c)/(d − c) allows us to prove the statement with respect to the standard interval [0, 1]. Pick α ∈ (0, 1] and a continuous mapping f : E →[0, α].Divide[0, α] in three subintervals of length (1/3)α: 1 1 2 2 I = 0, α , I = α, α I = α, α . 1 3 2 3 3 3 3 Now set 1 2 A = x ∈ E : f (x) ≤ α and B = x ∈ E : f (x) ≥ α . 3 3 These are disjoint, and because f ∈ C0(E) (the class of continuous maps on E)they are closed in E, hence in X. By Urysohn’s lemma there exists a continuous function gˆ : X →[0, 1] mapping A to 0 and B to 1. Then g1(x) = (α/3)gˆ(x), x ∈ X, is continuous and g1(x) = 0, ∀x ∈ A; ( ) = α , ∀ ∈ ; g1 x 3 x B ( ) ∈ , α , ∀ ∈ \ ( ∪ ). g1 x 0 3 x X A B Furthermore, for x ∈ E 1 2 0 ≤ f (x) − g (x) ≤ α − α = α. 1 3 3 When α = 1wehaveg1 : E →[0, 1/3]. Set 2 ( f − g ) = f : E → 0, . 1 1 3 Repeating the previous step for α = 2/3 gives the three subintervals 1 2 1 2 2 2 2 2 2 I = 0, · , I = · , , I = , , 1 3 3 2 3 3 3 3 3 3 and we redefine Appendix B: Urysohn’s Lemma and Tietze’s Theorem 443 1 2 2 2 A = x ∈ E : f (x) ≤ · and B = x ∈ E : f (x) ≥ . 1 3 3 1 3 still closed and disjoint in X. The Urysohn lemma gives a new continuous map gˆ : X →[0, 1] sending A to 0 and B to 1. The continuous map g2(x) = (1/3) · (2/3)gˆ(x) satisfies ( ) = , ∀ ∈ ; g2 x 0 x A 1 2 g2(x) = · , ∀x ∈ B; 3 3 ( ) ∈ , 1 · 2 , ∀ ∈ \ ( ∪ ); g2 x 0 3 3 x X A B and for every x ∈ E 2 2 0 ≤[f (x) − (g (x) + g (x))]≤ . 1 2 3 Next, consider 2 2 [ f − (g + g )]= f : E → 0, . 1 2 2 3 By induction, at the nth iteration we have a continuous map gn such that ( ) = , ∀ ∈ ; gn x 0 x A n−1 ( ) = 1 2 , ∀ ∈ ; gn x 3 3 x B n−1 ( ) ∈ , 1 · 2 , ∀ ∈ \ ( ∪ ); gn x 0 3 3 x X A B and n 2 n 0 ≤ f (x) − g (x) ≤ i 3 i=1 for every x ∈ E. Taking the limit as n →∞we finally obtain ∞ g(x) = gi (x), x ∈ E. i=1 The above turns into the required extension g : X →[0, 1], provided we show it ∞ ( ) ( ) = ( ) is continuous, i.e. i=1 gi x converges uniformly on X, and that g x f x for every x ∈ E. 444 Appendix B: Urysohn’s Lemma and Tietze’s Theorem Define Sn(x) = g1(x) + g2(x) + ...+ gn(x). For any n ∈ N the partial sum Sn is continuous, and whenever m > n m m |Sm (x) − Sn(x)|= gk (x) ≤ |gk (x)| = + = + k n 1 k n 1 − n m 1 m 2 k 1 1 2 − 2 ≤ = · 3 3 − 2 3 = + 3 3 1 k n 1 3 n m n = 2 − 2 ≤ 2 3 3 3 for every x ∈ X. Hence {Sn}n≥1 is a Cauchy sequence. Taking the limit as m →∞, we know lim Sm (x) = g(x) m→∞ so the above inequality reads 2 n |g(x) − S (x)|≤ , x ∈ X. n 3 Therefore {Sn}n≥1 converges uniformly on X to g(x), whence the continuity of g follows. Furthermore g(x) is bounded, since ∞ ∞ − 1 2 i 1 1 1 0 ≤ g(x) = gi (x) ≤ = · = 1. 3 3 3 1 − 2 i=1 i=1 3 At last, given x ∈ E, by construction 2 n f (x) − S (x) = f (x) − g (x) + ...+ g (x) ≤ . n 1 n 3 As n →∞we obtain f (x) − g(x) = 0, so f (x) = g(x) for every x ∈ E. Proof of 2. Now we have a continuous extension g : X →[0, 1] of f : E → [0, 1]. Since R is homeomorphic to (0, 1), it will suffice to exhibit a continuous extension φ of f with image φ(X) ⊆ (0, 1). Notice that by replacing f with (2 f − 1) and g with (2g − 1) we can prove 2. for the interval [−1, 1]. So consider F = g−1({−1, 1}) ⊆ X. Appendix B: Urysohn’s Lemma and Tietze’s Theorem 445 Clearly F is closed, and disjoint from E because g(E) ⊆ (−1, 1). Urysohn’s lemma guarantees there exists a continuous function h : X →[0, 1] such that h(F) ={0} and h(E) ={1}. The continuous map φ = h · g extends f , since for every x ∈ E φ(x) = h(x) · g(x) = g(x) = f (x); its image lies in (−1, 1), because for x ∈ F φ(x) = g(x) · h(x) = 0; and when x ∈/ F |φ(x)|=|g(x) · h(x)|≤|g(x)| < 1. List of Figures 2.1 Table of rational numbers … 26 2.2 Construction of Volterra’s set - the first three iterations … 34 3.1 Hierarchy of Borel classes … 76 IV.1 Construction of the Riemann integral of a function f (x) … 152 IV.2 Graph of y = (x) … 153 IV.3 Graph of Riemann’s function R(x) for n = 2…154 IV.4 Graph of Riemann’s function R(x) for n = 10 … 154 ( ) = 2 1 → + IV.5 Graph of f x x sin x as x 0 … 159 IV.6 Graph of fa,b(x), part of Volterra’s function … 160 IV.7 Graph of the Volterra function … 161 10.1 Graphs of y = min(x, x2) and y = max(x, x2) … 251 10.2 Graph of the Vitali-Cantor map … 271 12.1 Example of continuous extension … 332 − 1 12.2 Graph of f (x) = e x on x > 0 and f (x) = 0onx ≤ 0…333 ( ) = f (x) 12.3 Graph of F x f (x)+ f (1−x) … 336 ( ) = g(x) 12.4 Graph of G x g(x)+g(−1−x) … 336 12.5 Graph of bump function φn … 337 14.1 Density and distribution functions of a uniform RV Ua,b … 421 14.2 Density function of a negative-exponential RV … 423 14.3 Density function of a normal RV … 424 © The Editor(s) (if applicable) and The Author(s), under exclusive 447 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 List of Symbols A ⊂ BAis a proper subset of B, 19 A ⊆ BAis a subset of B, 4 A(C) Algebra generated by a family C, 124 A+ Algebra generated by pairwise-disjoint sets of a semi-algebra, 125 B(n, p) Binomial random variable of order n and parameter p, 413 B(R) Borel σ-algebra over R, 142 B(A) Boundary of a set A, 22 C Cantor set, 40 c Cardinality of the continuum, 27 ℵ0 Cardinality of the set N, 27 ϕA Characteristic function of a set A, 67 I Class of all real intervals, 122 0 R 1 Class of closed subsets of , 56 0 0 α Class of countable intersections of sets in β<α β, 56 0 0 β Class of countable intersections of sets in β−1, 56 0 0 α Class of countable unions of sets in β<α β, 58 0 0 β Class of countable unions of sets in β−1, 56 M +( , S) S 0 X Class of non-negative -measurable simple maps, 256 0 R 1 Class of open subsets of , 56 Apj(I) Class of Peano-Jordan measurable subsets of R, 172 I˜ Class of real right-closed intervals, 122 M0(X, S) Class of S-measurable simple maps, 256 N Class of subsets of X with zero outer measure, 204 B¯ (x, r) Closed ball, 30 A¯ Closure of a set A, 4 © The Editor(s) (if applicable) and The Author(s), under exclusive 449 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 450 List of Symbols G (I ) Collection of improperly integrable maps on a bounded interval I , 339 G (I ∗) Collection of improperly integrable maps on unbounded interval I , 344 R(I ) Collection of Riemann integrable, bounded maps, 324 Ac Complement set to A, 22 C( f ) Continuity set of f , 64 Fσ,δ Countable intersections of Fσ sets, 57 Gδ Countable intersections of open sets of R, 55 Fσ Countable unions of closed sets of R, 57 Gδ,σ Countable unions of Gδ sets, 55 F (x) = p Cumulative distribution function of a discrete RV X, X xi ∂ A Frontier of a set A, 7 inf Greatest lower bound or infimum, 108 ∼ has the same distribution as, 426 ⇔ If and only if, 5 ⇒ Implies, 4 ( ) = ( )( ) F x [a,x] f t dt Integral function, 349 ∈ M ( , S) X fdm Integral over X of map X with respect to m, 290 ∈ M +( , S) X sdm Integral over X of a simple map 0 X with respect to m, 286 ◦ A Interior of a set A, 5 , ∩ Intersection, 5 R \ Q Irrational numbers, 25 sup Least upper bound or supremum, 63 ( )( ) [ , ] [a,b] f x dx Lebesgue integral of f on a b , 323 ∗ Lebesgue outer measure, 218 L Lebesgue σ-algebra of measurable subsets of X, 202 L (R) Lebesgue σ-algebra on R, 218 m F Lebesgue-Stieltjes measure generated by F, 234 s( f ) Lower Riemann integral of a bounded map f : [a, b] → R, 324 ( , ) s f Pn Lower sum over partition Pn, 324 ( ) = E X i∈I pi xi Mean value of a discrete RV X, 402 E(X) = R xf(x)(dx) Mean value of an absolutely continuous RV X, 402 (X, S) Measurable space, 213 (X, S, m) Measure space, 213 (X, d) Metric space, 29 M Monotone class, 134 M (C) Monotone class generated by C, 134 n R | x | f (x)dx nth absolute moment of an absolutely continuous RV, 403 ∞ | |n i=1 xi pi nth absolute moment of a discrete RV, 403 n R(x − m) f (x)dx nth central moment of an absolutely continuous RV, 403 ∞ ( − )n i=1 xi m pi nth central moment of a discrete RV, 403 n ( ) R x f x dx nth moment of an absolutely continuous RV, 403 ∞ n i=1 xi pi nth moment of a discrete RV, 403 f − Negative part of a map, 248 v ( ) = x ( ) − x VNa f Negative total variation function of f , 365 b( , ) [ , ] Va P− f Negative variation of f over a b for partition P, 364 Ux Neighbourhood of a point x, 4 N(m, σ2) Normal RV, 426 First uncountable ordinal, 55 452 List of Symbols B(x, r) Open ball, 30 ω( f, x) Oscillation of f at x, 63 ω( f, E) Oscillation of f over E, 63 ∗ m (A) Outer measure of a set A, 197 Pairwise-disjoint union, 59 m pj(A) Peano-Jordan measure of a set A, 172 = λk e−λ pk k! Poisson RV, 417 f + Positive part of a map, 248 Q+ Positive rationals, 25 v ( ) = x ( ) + x VPa f Positive total variation function of f , 365 b( , ) [ , ] Va P+ f Positive variation of f over a b for partition P, 364 P(X) Power set of X: collection of subsets of X, 123 f −1(E) Pre-image {x ∈ X : f (x) ∈ E}, 74 {pi }i≥1 Probability distribution of a discrete RV, 399 PX Probability distribution of a RV X, 274 P() Probability of certain event, 276 P(X = x) Probability of elementary event, 279 P(A) Probability of event A, 276 pi Probability of value xi for a discrete RV, 399 (, S(), P) Probability space, 274 P(X ∈ B) Probability the variable X takes values in B, 274 X : → R Random variable (RV), 274 Q Rational numbers, 25 ∗ Restriction of measure to L (R) or B(R), 222 b ( ) [ , ] a f x dx Riemann integral of f on a b , 323 Sample space, 274 {An}n≥1 Sequence of sets with nth term An, 49 m F Set function induced by map F, 234 A \ B Set of elements of A that do not belong to B, 3 R¯ Set of extended reals, 11 Z Set of integers, 28 N Set of natural numbers, 19 R Set of real numbers, 3 S() σ-algebra of events, 274 S(A) σ-algebra generated by an algebra, 138 S(C) σ-algebra generated by C, 131 s(x) Simple functions, 106 R([a, b]) Space of maps in R(I ), 328 ∞(R) R Cc Space of compactly supported smooth maps on , 338 C([a, b]) Space of continuous maps on [a, b], 331 Cc(R) Space of continuous maps with compact support on R, 337 List of Symbols 453 L1(X, S, m) or L1 Space of m-summable functions, 304 C∞([a, b]) Space of smooth functions on [a, b], 336 C∞(R) Space of smooth functions on R, 333 ([ , ]) [ , ] √L1 a b Space of summable functions on a b , 326 Var(X) or σ(X) Standard deviation of a RV X, 403 N(0, 1) Standard normal RV, 427 C ∩ Y Subsets of Y contained in C, 145 supp( f ) Support of a map, 337 (X, T ) Topological space, 3 T Topology, 3 b( ) [ , ] VNa f Total negative variation of f on a b , 364 b( ) [ , ] VPa f Total positive variation of f on a b , 364 v( ) = x ( ) x Va f Total variation function of f , 365 b( ) [ , ] Va f Total variation of f over a b , 355 V ( f ) Total variation of f over R, 355 U [a,b] Uniform RV, 420 , ∪ Union, 5 S( f ) Upper Riemann integral of a bounded map f : [a, b] → R, 324 ( , ) S f Pn Upper sum over partition Pn, 324 ( ) = ∞ ( − )2 Var X i=1 xi m pi Variance of a discrete RV X, 403 Var(X) or σ2(X) Variance of a RV X, 403 2 Var(X) = R(x − m) f (x)dx Variance of an absolutely continuous RV X, 403 b( , ) [ , ] Va P f Variation of f over a b with respect to partition P, 355 x ∈ Axbelongs to the set A, 4 x ∈/ Axdoes not belong to the set A, 8 References 1. Apostol Tom M. (1974) - Mathematical Analysis - second ed. - Addison Wesley - Massachusetts 2. Athreya Krishna B., Lahiri Soumendra N. (2006) - Measure Theory and Probability Theory - Springer - New York 3. Bartle Robert G. (2001) - Elements of Real Analysis - second ed. - John Wiley and Sons - New York 4. Billingsley Patrick (1995) - Probability and Measure - third ed. - John Wiley and Sons - New York 5. Bottazzini Umberto (1981) - Il Calcolo sublime: storia dell’analisi matematica da Euler a Weierstrass - Editore Boringhieri - Torino 6. Bottazzini Umberto (1988) - L’aritmetizzazione dell’analisi - vol. 2 - tomo 2 - p.953 - Storia della Scienza Moderna e Contemporanea - Editore Utet - Torino 7. Bottazzini Umberto (1988) - Insiemi di punti e numeri transfiniti - vol. 3 - tomo 1 - p.55 - Storia della Scienza Moderna e Contemporanea - Editore Utet - Torino 8. Bottazzini Umberto (1988) - L’analisi funzionale, “nuova branca della matematica” -vol.3 - tomo 1 - pag. 535 - Storia della Scienza Moderna e Contemporanea - Editore Utet - Torino 9. Carathéodory Constantin (1918) - Vorlesungen über reelle Funktionen - Leipzig - Berlino 10. Carathéodory Constantin (1956) - Algebraic Theory of Measure and Integration -Chelsea Publishing Company - New York 11. Dall’Aglio Giorgio (2003) - Calcolo delle Probabilità - third ed. - Zanichelli - Bologna 12. De Finetti Bruno (1990) - Theory of Probability - A Critical Introductory Treatment -Vol.1- John Wiley and Sons - New York 13. De Finetti Bruno (1990) - Theory of Probability - A Critical Introductory Treatment -Vol.2- John Wiley and Sons - New York 14. Geymonat Ludovico (2008) - Storia e Filosofia dell’Analisi Infinitesimale - Bollati Boringhieri -Editore-Torino 15. Giusti Enrico (1992) - Analisi Matematica 1 - Editore Boringhieri - Torino 16. Giusti Enrico (1992) - Analisi Matematica 2 - Editore Boringhieri - Torino 17. Goffman Casper (1970) - Real Functions - The Prindle, Weber and Schmidt - Boston 18. Kechris Alexnder S. (1995) - Classical Descriptive Set Theory - Springer-Verlag - New York 19. Kuratowski Kazimierz (1966) - Topology - vol. 1 - Academic Press - New York 20. Kurtz Douglas S., Swartz Charles W.(2004) - Theories of Integration (The Integrals of Riemann, Lebesgue, Henstock-Kurzweil and Mcshane) - World Scientific Publishing - Singapore © The Editor(s) (if applicable) and The Author(s), under exclusive 455 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 456 References 21. Lebesgue Henri (1902) - Intégrale, Longueur, Aire - Annali di Matematica Pura e Applicata - serie III - tomo VII - p.231 - Milano 22. Lebesgue Henri (1904) - Leçons sur l’intégration et la recherche des fonctions primitives - Gauthier-Villars - Parigi 23. Natanson Isidor Pavlovi˘c (1955) - Theory of Functions of a Real Variable -vol.1-Frederick Ungar Publlishing Co. - New York 24. Natanson Isidor Pavlovi˘c (1960) - Theory of Functions of a Real Variable -vol.2-Frederick Ungar Publlishing Co. - New York 25. Piccolo Domenico (1998) - Statistica - Il Mulino - Bologna 26. Sierpi´nski Wacław (1924)- Sur une propriété des ensembles ambigus - Fundamenta Mathe- maticae vol. VI - Varsavia 27. Sierpi´nski Wacław (1934) - Introduction to General Topology - The University of Toronto Press - Toronto 28. Vinti Calogero (1988) - Lezioni sulla Teoria dell’Integrazione - Galeno Editrice - Perugia 29. Yeh James J. (2006) - Real Analysis (Theory of Measure and Integration) - second ed. - World Scientific Publishing - Singapore Index A Borel–Cantelli lemma, 258 Absolutely continuous map Borel functions - a.e. zero derivative, 387 - classification, 105 - bounded variation, 380 -ofclass2,115 - definition, 376 Borel measurable functions - differentiability, 383 - definition, 242 - operations, 378 Borel sets in R - summability, 383 - measurability, 219 Absolutely continuous measure Boundary - definition, 314 - definition, 22 Absolutely perfect set - properties, 22 - structure in R, 38 Bounded nowhere dense perfect set Additive class, 69 - structure in R, 38 A.e. equality of maps - equivalence relation, 218 Algebra of sets C - definition, 123 C([a, b]) - finite-cofinite, 128 - density in L1([a, b]), 331 - generated by a semi-algebra, 124 Cantor - generated by an arbitrary family, 127 R + - intersection theorem for , 19 - generated by C , 125 Cantor set, 39 - minimal, 124 Carathéodory Almost uniform convergence - extension theorem, 204 - definition, 260 Cauchy criterion Ambiguous set - for improper integrals over I , 339 - definition, 75 - for improper integrals over I ∗, 344 -ofclassα, 80 ∞(R) Cc -structure,80 - density in L1(R), 338 Cc(R) - definition, 337 B - density in L1(R), 337 Baire functions Chebyshev - cardinality, 103 - inequality, 292 - definition of Baire classes, 90 C∞([a, b]) - operations, 90 - density in L1([a, b]), 336 © The Editor(s) (if applicable) and The Author(s), under exclusive 457 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 458 Index C∞(R) -closed,13 - definition, 333 - definition, 13 0 Class α -iterated,13 - prominent relationships, 76 - properties, 16 - properties, 75 Diameter of a set 0 Class α - definition, 433 - properties, 70 Dirichlet function 0 Class α - Baire class, 101 - properties, 70 Distance from a set Closed set - uniform continuity, 108 - characterisation, 13 - definition, 3 - derived set, 18 E - intersection with compact set, 22 Elementary event - properties, 3 - probability, 279 Closed set with empty interior Elementary probability of absolutely contin- - countable union, 47 uous RVs Closed subset in R - definition, 401 -structure,22 Expected value Closure - conditional - definition, 4 - definition, 409 Closure and interior - of product of independent discrete RVs, - remarkable properties, 6 409 Comeagre set - definition, 48 Compact set F -inR, 21 Family of functions - in metric spaces, 435 - tight - intersection with closed set, 22 - definition, 318 Composite maps - uniformly summable - Baire class, 93 - definition, 316 Continuity of measure Finite Borel measure - from above, 195 - regularity conditions, 224 -frombelow,194 Finitely additive measure Continuous map - definition, 168 - Borel measurability, 242 Finite subcover Cumulative distribution function - definition, 20 - absolutely continuous Fourier representability - definition, 401 - sufficient conditions, 85 - definition, 276 Frontier - discrete - definition, 7 - definition, 399 Fσ set - properties, 276 - Baire category, 62 - uniform continuity, 278 - closure under unions/intersections, 62 - complement, 65 - difference of closed sets in R, 60 D -interval,58 Dense set, 25 - open subset of R, 60 - characterisation, 25 - σ-compact set in R, 59 -oftypeGδ - union of intervals, 59 - Baire category, 62 Function - relationship with nowhere dense set, 28 - absolute continuity vs. bounded varia- Derived set tion, 390 Index 459 - absolute continuity vs. monotonicity, Integral function 391 - absolute continuity, 377 + - in Baire class B1 Integral of function in M (X, S) - Baire category of discontinuity set, -finite,301 99 - properties, 291 - nature of the discontinuity set, 96 - vanishing condition, 299 - of absolutely continuous RV - w.r.t. density measure, 297 - probability density, 401 Integral of L1 functions - with bounded variation - vanishing condition of f ∈ L1, 308 M +( , S) - definition, 355 Integral of simple map in 0 X - differentiability, 375 - definition, 286 - operations, 361 - properties, 287 - relationships among f,v,v+,v−, - remarkable inequality, 289 366 Integration - summability, 375 -byparts,392 Functions in G (I ∗), 344 - by substitution, 397 - measurability, 344 - lemma to substitution formula, 393 Functions in G (I ), 339 Interior - measurability, 339 - definition, 5 Interval - Baire category, 49 G Gδ set - cardinality, 66 L - closed subset of R, 59 L1 function - closure under unions/intersections, 62 - properties, 304 - complement, 65 - properties of the integral, 305 - difference of open sets in R, 60 L1(I ) and G (I ) -interval,58 - relationship, 340 ∗ ∗ Generalised Cantor set, 328 L1(I ) and G (I ) - relationship, 345 L1([a, b]) I - completeness, 330 Improper integral Lebesgue integral - of dominated map, 340, 344 - additivity w.r.t. integration domain, 309 - over bounded interval, 338 Lebesgue measurable function - over unbounded interval, 344 - definition, 242 Inequality Lebesgue measure - Bienaymé–Chebyshev, 407 - outer regularity, 222 - Chebyshev, 292 Lebesgue-Stieltjes measure -Markov,406 - generated by F, 237 -Schwarz,405 - space, 237 Inner Peano-Jordan measure Lemma - definition, 171 - Borel–Cantelli, 258 Inner regular measure - Fatou, 298 - definition, 224 - Urysohn, 443 Integrability - Vitali covering, 369 - Riemann Limit point - PJ measurability, 181 - characterisation, 10 Integrable function - definition, 10 - definition, 303 Littlewood Integral calculus - first principle, 270 -1st fundamental theorem, 383 - second principle, 262 -2nd fundamental theorem, 389 - third principle, 266 460 Index M - definition, 185 Maps in M +(X, S) - continuity from above, 195 - comparing integrals, 302 - continuity from below, 194 Meagre sets - definition of restriction, 208 - countable unions, 48 - σ-finite, 207 - definition, 48 - totally finite, 208 - subsets, 48 Measure on a semi-algebra Measurability - uniqueness of extension, 192 - Carathéodory, 202 Measure space, 213 - equivalent definitions, 202 - complete - of Dirichlet-like maps, 253 - definition, 213 - Peano-Jordan - completeness, 214 - characterisation, 178 Measure with density f ∈ M +(X, S) - Riemann integrability, 181 - with respect to reference measure, 296 ∈ M +( , S) Measurable cover Measure with density s 0 X -ofaset,211 - with respect to reference measure, 288 Measurable function Metric space - a.e. bounded - bounded set - approximation by continuous maps, - definition, 433 266 - compact - characterisation, 241 - completeness and total bounded- - definition, 241 ness, 439 -definitionofintegralonS-set, 290 - sequential compactness, 437 -withvaluesin[0, ∞] - complete - characterisation via sequences of - Cantor’s intersection theorem, 441 simple maps, 263 - compactness, 440 - definition of integral, 290 - definition, 31 - integral as limit integral of simple - Heine–Pincherle–Borel theorem, maps, 291 440 -withvaluesinR¯ - complete and totally bounded - characterisation, 249 - compactness, 439 - characterisation via sequences of - convergent sequences simple maps, 263 - Cauchy condition, 31 - comparison, 245 - countable compactness - definition, 243, 263 - definition, 436 - definition of negative part, 248 - definition, 29 - definition of positive part, 248 - definition of ball, 30 - equality m-a.e., 252 - definition of limit of sequence, 31 - equality m-a.e. of S¯ and S- - discrete measurable maps, 252 - definition, 434 - equivalent definitions, 243 - separable - lim inf and lim sup of sequences, - definition, 31 251 - sequentially compact - min and max, 250 - compactness, 437 - operations, 246 - definition, 435 - sup and inf of sequences, 249 - sequential compactness versus total Measurable kernel boundedness, 435 -ofaset,211 - sequential versus countable compact- Measurable set ness, 436 - approximation, 216 - totally bounded Measurable space, 213 - separability, 434 Measure - totally bounded set - completely additive - boundedness, 433 Index 461 - definition, 433 - definition, 197 Moment of RV - infimum of measures of disjoint covers, - definition, 403 198 - existence of lower-order moments, 404 - properties, 198 Monotone class Outer Peano-Jordan measure - definition, 134 - definition, 171 - Halmos’s theorem, 136 Outer regular measure - minimal, 135 - definition, 224 Monotone class and algebra - σ-algebra, 136 Monotone function P - differentiability a.e., 371 Peano-Jordan measurable set - discontinuity, 368 - definition, 172 Multiplicative class, 69 Peano-Jordan measure - definition, 172 - finite additivity, 180 N - properties, 172 Negative total variation function Perfect set - properties, 367 - cardinality, 441 - definition, 29 Negligible set Point - definition, 213 - boundary, 8 Neighbourhood -interior,6 - definition, 4 - isolated Non-Lebesgue measurable set, 233 - characterisation, 10 Non-negligible nowhere dense set - definition, 10 - Smith’s example, 35 Pointwise convergence - Volterra’s example, 33 - of sequences of functions Nowhere dense set, 25 - definition, 89 - characterisation, 27 Pointwise convergence a.e. - definition, 24 - of sequences of functions - relationship with dense set, 28 - definition, 256 - second characterisation, 27 - measurability of limit, 256 - subsets, 28 - properties, 261 - relation to a.u. convergence, 262 Pointwise-convergent sequences O - Borel class, 114 Open cover Pointwise-limit function - definition, 20 - Borel class, 110 Open dense set - pre-image, 107 - countable intersections, 45 Positive total variation function Open set in R - properties, 367 -structure,12 Principle Operator - of transfinite induction, 68, 69 -closure Probability density function - properties, 4, 7 - definition, 401 - frontier Probability distribution - properties, 8 - definition, 274 -interior - properties, 7 Oscillation of a real function R - definition, 63 R Outer measure - Baire category, 50 - comparison, 200 R([a, b]) 462 Index - incompleteness, 329 - uniformly summable and tight, 318 Random variable Sequences of measurable functions - almost constant, 410 - absolute convergence a.e., 259 - binomial, 411 - complement of convergence set - continuous - measurability, 257 - definition, 398 - convergence set - definition, 274 - measurability, 257 - difference, 275 - pointwise convergence a.e., 257, 259 - discrete Set - definition, 398 - boundary, 22 - independent, 409 - countable cover, 186 - function of, 274 - dense in itself, 25 - integral of measurable functions of, 402 - inner measure of interior, 177 - linearity, 407 - measurable cover, 211 - negative-exponential, 421 - measurable kernel, 211 - normal, 423 - of first species - Poisson, 414 - definition, 14 - product, 275 - negligible, 32 - quotient, 275 - nowhere dense, 32 -sum,275 - of isolated points - uniform, 420 - cardinality, 17 Real function - definition, 13 - continuous on a residual set, 67 - of second category - definition of oscillation, 63 - definition, 48 - discontinuity set, 64 - of second species - oscillation at continuous points, 63 - definition, 14 Real-valued simple function - outer measure of closure, 177 - Borel class, 106, 112 Set function - characterisation of Borel-class, 106 - completely additive, 167 - definition, 106 - completely subadditive, 168 Residual set - definition, 167 - characterisation, 49 - extension, 171 - countable intersections, 49 - finitely additive, 167 - definition, 48 - induced by increasing map, 234 - in complete metric space - induced map, 237 - cardinality, 50 - monotone, 167 Residual subset in R - over an algebra - Baire category, 50 -1st case of countable additivity, 193 Restriction - properties, 193 - Borel class, 105 - σ-finite, 168 Riemann integrable map - totally finite, 168 - summability, 324 σ-algebra L ¯ R-valued simple function - approximation of sets, 220–222 - definition, 253 σ-algebra of sets - measurability, 256 -Borel - operations, 254 - cardinality, 229 - equivalent ways to generate, 144 - generated by closed subsets of R, S 143 Semi-algebra of sets -onR, 142 - definition, 122 - definition, 131 Sequences of functions - equivalent generating processes, 141 - uniformly summable, 316 - generated by A+, 139 Index 463 - generated by an algebra, 138 - Vitali’s convergence, 317 - generated by an arbitrary family, 140 - extension, 319 - Lebesgue L , 206 Topological space -structure,214 - definition, 3 - minimal, 131 Topology - monotone class, 135 - definition, 3 - monotone class and algebra, 136 Total variation - restriction, 145 - additivity, 360 σ-finite measure on an algebra A -negative - uniqueness of extension to S(A), 208 - definition, 363 Simple function with values in [0, ∞] - positive -definitionofintegral,286 - definition, 363 - integral over S-set, 286 Total variation function - properties of integral, 287 - definition, 365 Singular map - properties, 365 b( ), b( ) - definition, 391 Total variation, Va f VPa f and b( ) Summability VNa f - equivalent criteria, 314 - relationships, 364 Summable function Translation in L (R) - definition, 303 - L-measurability, 227 Support - definition, 337 U Uniform convergence T - Baire class stability, 95 Theorem - Borel class stability -Baire,49 - necessary condition, 111 - Bolzano–Weierstraß, 18 - of restrictions, 105 - in metric spaces, 435 - sufficient condition, 111 - bounded convergence, 313 - of sequences of functions - dominated convergence (Lebesgue), - definition, 89 310 - pre-image, 109 - for series, 312 Urysohn function - Hahn-Kolmogorov - example on a metric space, 445 - uniqueness of extension of a σ-finite measure, 208 - Heine–Pincherle–Borel in R, 20 V - Jordan, 367 Variance of random variable - Lebesgue decomposition, 391 - as function of moments, 408 - Lebesgue–Hausdorff, 116 - vanishing, 410 -Lusin,266 Vitali covering - monotone-class (Halmos), 136 - definition, 369 - monotone convergence (Beppo Levi), Vitali’s covering lemma, 369 293 Vitali set, 232 - consequences, 295 - measurable subsets, 233 - reduction, 78 - Riesz-Fischer, 330 - separation, 79 Z - Severini–Egorov, 262 Zero-measure set - Tietze’s extension, 445 - definition, 213