Walter Rudin 224)
- The components of f are the real functions f, \.\.\., f,, defined by \(24 \(00 =3 fitow \(xe£\), (Walter Rudin 224)
- ForxeE, 1 *Walter Rudin 224)
- f'0Oh =% { S \(D,f,\.\)\(x\)h,\.} u,\. = (Walter Rudin 225)
- Associate with each x € E a vector, the so-called “gradient” of f at x, defined by \(4 \(V\) = 3\.\(Dufe (Walter Rudin 226)
- Hence \(38\) gives \(39\) L S+ \) —\) = \(V\) \(%\) u\. t—0 t (Walter Rudin 226)
- Iffand x are fixed, but u varies, then \(39\) shows that \(D,f\)\(x\) attains its maximum when u is a positive scalar multiple of \(V/\)\(x\)\. (Walter Rudin 227)
- Corollary If, in addition, £'\(x\) = 0 for all x € E, then { is constant\. (Walter Rudin 227)
- The limit in \(39\) is usually called the directional derivative off at x, in the direction of the unit vector u, and may be denoted by \(D,f\)\(x\)\. (Walter Rudin 227)
- 9\.21 Theorem Suppose f maps an open set E = R" into R™\. Then fe ¢'\(E\) if and only ifthe partial derivatives D,f, exist and are continuous on E (Walter Rudin 228)
- 9\.20 Definition A differentiable mapping f of an open set E < R" into R" is said to be continuously differentiable in E if ' is a continuous mapping of E into L\(R", R™\)\. (Walter Rudin 228)
- 9\.22 Definition Let X be a metric space, with metric d\. If ¢ maps X into X and if there is a number ¢ < 1 such that 43\) d\(o\(x\), ¢\(»\)\) < cd\(x, y\) for all x, y € X, then ¢ is said to be a contraction of X into X\. (Walter Rudin 229)
- 9\.23 Theorem If X is a complete metric space, and if ¢ is a contraction of X into X, then there exists one and only one x € X such that ¢\(x\) = x\. (Walter Rudin 229)
- 9\.24 Theorem Suppose fis a €'-mapping of an open set E = R" into R", f'\(a\) is invertible for some a € E, and b =f\(a\)\. Then \(@\) there exist open sets U and V in R" such that ac U, be V, f is one-tooneon U, and f\(U\) =V, \(b\) if g is the inverse of £ [which exists, by \(a\)], defined in V by gfx\) =x \(xel\), then g e €'\(V\)\. (Walter Rudin 230)
- The system of n equations Vi=f\(X1, \.00y Xp\) \(1**Walter Rudin 230)
- if we restrict x and y to small enough neighborhoods of a and b; the solutions are unique and continuously differentiable\. (Walter Rudin 230)
- \) Proof (Walter Rudin 230)
- This completes the proof\. (Walter Rudin 232)
- In other words, f is an open mapping of E into R"\. (Walter Rudin 232)
- 9\.25 Theorem I[ff is a €'-mapping of an open set E = R" into R" and if £'\(x\) is invertible for every x € E, then £\(W\) is an open subset of R" for every open set WcE (Walter Rudin 232)
- This may be expressed by saying that f is locally one-to-one in E\. But f need not be 1-1 in E under these circumstances\. (Walter Rudin 232)
- 9\.27 Theorem If A e L\(R"\*™R"\) and if A, is invertible, then there corresponds to every k € R™ a unique h € R" such that A\(h, k\) = 0\. This h can be computedfrom k by the formula \(55\) h=—\(4,\)""'4,k Proof By \(54\), A\(h, k\) =0 if and only if Ah+ Ak=0, which is the same as \(55\) when A, is invertible\. (Walter Rudin 233)
- A\(h, k\) = 0 can be solved \(uniquely\) for h if k is given, and that the solution h is a linear function of k\. (Walter Rudin 233)
- The assumption that A4,is invertible means that the n by n matrix [D1f1 anl} Dl/;l an;: evaluated at \(a, b\) defines an invertible linear operator in R"; (Walter Rudin 234)
- This is equivalent to \(58\), and completes the proof\. (Walter Rudin 236)
- Definition A family & ofsets is called a ring if A € # and B € # implies D AUBeX, A-Be®\. SinceA nB=A— \(A — B\), we also have A n Be # if # 1s a ring\. A ring Z is called a o-ring if \(2\) J4,e2 n=1 (Walter Rudin 310)
- ﬁAne\.% n=1 (Walter Rudin 310)
- if & is a o-ring\. (Walter Rudin 310)
- We say that ¢ is a set function defined on Z if ¢ assigns to every A € # a number ¢\(A\) of the extended real number system\. ¢ is additive if A n B =0 implies A3\) P\(4 v B\) = ¢\(4\) + ¢\(B\), and ¢ is countably additive if A; N A; =0 \(i # j\) implies @ 6 \(04\) =3o4, (Walter Rudin 310)
- the rearrangement theorem shows that the right side of \(4\) converges absolutely if it converges at all; (Walter Rudin 310)
- $\(A; VU 4;\) + ¢\(4; N A;\) = ¢\(4,\) + $\(4;\)\. (Walter Rudin 311)
- 6\(4;\) < B\(4,\)\. (Walter Rudin 311)
- nonnegative additive set functions are often called monotonic\. (Walter Rudin 311)
- ¢\(4 — B\) = ¢\(A4\) — ¢\(B\) if B< 4, and |\(¢B\)| < + oo\. (Walter Rudin 311)
- 11\.3 Theorem Suppose ¢ is countably additive on a ring \. Suppose A, € R n=1213\.\),4,cA,c Ay, AeR, and A= QA,,\. Then, as n — 0, P\(4,\) = P\(A4\)\. (Walter Rudin 311)
- If A is the union of a finite number of intervals, A4 is said to be an elementary set\. (Walter Rudin 312)
- If 7 is an interval, we define m\(d\) = 11 6: =\), (Walter Rudin 312)
- We let & denote the family of all elementary subsets of R?\. (Walter Rudin 312)
- &is aring, but not a o-ring\. \(13\) If A € &, then A is the union of a finite number of disjoint intervals\. (Walter Rudin 312)
- A nonnegative additive set function ¢ defined on & is said to be regular if the following is true: To every 4 € & and to every & > 0 there exist sets F € &, G € & such that Fis closed, G is open, F < 4 < G, and \(16\) $\(G\) — & < P\(4\) < ¢\(F\) +&\. (Walter Rudin 312)
- Our next objective is to show that every regular set function on & can be extended to a countably additive set function on a o-ring which contains &\. (Walter Rudin 313)
- 11\.7 Definition Let u be additive, regular, nonnegative, and finite on &\. Consider countable coverings of any set £ = R? by open elementary sets A4,: Ec OIA,,\. Define " \(17\) K\*\(E\) = inf3u\(d,\), (Walter Rudin 313)
- the inf being taken over all countable coverings of E by open elementary sets\. u\*\(E\) is called the outer measure of E, corresponding to u\. (Walter Rudin 313)
- It is clear that u\*\(E\) = O for all E and that \(18\) P¥\(Ey\) < p\*\(E\)\) if E, c E,\. (Walter Rudin 313)
- For every A € &, u\*\(A\) = u\(A\)\. (Walter Rudin 313)
- IfE=|JE,, then 1 \(19\) p\*\(E\) < Z,"\*\(E"\)' (Walter Rudin 313)
- u\* is an extension of u from & to the family of all subsets of RP\. (Walter Rudin 313)
- The property \(19\) is called subadditivity\. (Walter Rudin 313)
- Choose A € & and ¢ > 0\. The regularity of u shows that 4 is contained in an open elementary set G such that u\(G\) < u\(A4\) + e\. Since p\*\(4\) < u\(G\) and since ¢ was arbitrary, we have (Walter Rudin 313)
- p\*\(A\) < u\(A\)\. The definition of u\* shows that there is a sequence {4,} of open elementary sets whose union contains A4, such that Zl W\(A,\) < w\*\(A\) +e\. (Walter Rudin 313)
- The regularity of u shows that A contains a closed elementary set F such that u\(F\) > u\(A\) — ¢; and since F is compact, we have F [ Al U otV AN for some N\. Hence N pA\) S p\(F\)+e*Walter Rudin 314)
- S\(A, By=\(4 — B\)u \(B — A\), (Walter Rudin 314)
- d\(A4, B\) = u\*\(S\(4, B\)\)\. (Walter Rudin 314)
- We write 4, A4 if limd\(4, A,\) =0\. (Walter Rudin 314)
- If there is a sequence {4,} of elementary sets such that 4, —» A4, we say that A is finitely u-measurable and write A € M\(u\)\. (Walter Rudin 314)
- If A is the union of a countable collection of finitely u-measurable sets, we say that A is u-measurable and write A € M\(y\)\. (Walter Rudin 314)
- except that d\(4, B\) = 0 does not imply 4 = B\. (Walter Rudin 315)
- For instance, if @ =m, A is countable, and B is empty, we have d\(A, B\) = m\*\(A\) = 0; (Walter Rudin 315)
- This extended set function is called a measure\. The special case u = m is called the Lebesgue measure on RP, (Walter Rudin 317)
- We now replace u\*\(4\) by u\(4\) if 4 € M\(u\)\. Thus u, originally only defined on &, is extended to a countably additive set function on the o-ring IM\(u\)\. (Walter Rudin 317)
- \(@\) If A is open, then 4 € M\(u\)\. For every open set in R? is the union of a countable collection of open intervals\. To see this, it is sufficient to construct a countable base whose members are open intervals\. By taking complements,it follows that every closed set is in M\(p\)\. (Walter Rudin 318)
- If A e M\(u\) and ¢ > 0, there exist sets F and G such that FcAcQgG, F is closed, G is open, and \(39\) WG — A\) < ¢, WA F\) Walter Rudin 318)
- We say that E is a Borel set if E can be obtained by a countable number of operations, starting from open sets, each operation consisting in taking unions, intersections, or complements\. (Walter Rudin 318)
- The collection # ofall Borel sets in RP is a g-ring; in fact, it is the smallest g-ring which contains all open sets\. By Remark \(a\), E € M\(u\) if E € AZ (Walter Rudin 318)
- If A € M\(u\), there exist Borel sets F and G such that Fc 4 < G, and \(40\) WG —A\) =4 -F\)=0\. This follows from \(b\) if we take ¢ = 1/n and let n — c0\. (Walter Rudin 318)
- Since A = F U \(A — F\), we see that every 4 € M\(u\) is the union of a Borel set and a set of measure zero\. (Walter Rudin 318)
- But the sets of measure zero [that is, the sets E for which u\*\(E\) = 0] may be different for different ws\. (Walter Rudin 318)
- But there are uncountable \(in fact, perfect\) sets of measure zero\. The Cantor set may be taken as an example: (Walter Rudin 318)
- 11\.28 Lebesgue’s monotone convergence theorem Suppose E € M\. Let {f,} be a sequence of measurable functions such that \(64\) 0Walter Rudin 327)
- 11\.30 Theorem Suppose E € M\. If{f,} is a sequence of nonnegative measurable functions and 16\) f\)=3fi® \(e, then Proof fEfd# =n§1 fEﬁ; The partial sums of \(76\) form a monotonically increasing sequence\. (Walter Rudin 329)
- 11\.31 Fatou’s theorem Suppose Ee€M\. If {f,} is a sequence of nonnegative measurable functions and f\(x\) =liminff\(x\) \(x€kE\), n— oo then 17 [ fdu Walter Rudin 329)
- 11\.32 Lebesgue’s dominated convergence theorem Suppose E € M\. Let {f,} be a sequence of measurable functions such that \(82\) fx\) 2f\(x\) \(xeE\) as n oo\. If there exists a function g € £\(u\) on E, such that \(83\) 4] Walter Rudin 330)
- Corollary If W\(E\) < + o0, {f,} is uniformly bounded on E, andf,\(x\) —f\(x\) on E, then \(84\) holds\. (Walter Rudin 331)
- Suppose f is bounded on [a, b\)\. Then f € & on [a, b] if and only iff is continuous almost everywhere on [a, b]\. (Walter Rudin 332)
- Conversely, if F is differentiable at every point of [a, b] \(“‘almost everywhere” is not good enough here!\) and if F’' € & on [a, b], then F\(x\) — F\(a\) = f F\(\) \(aWalter Rudin 333)
- Iffe % on [a, b], and \(96\) Foy=[fdt \(asxsb\), then F'\(x\) = f\(x\) almost everywhere on [a, b]\. (Walter Rudin 333)