Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S

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1 Riemnn-Stieltjes Polnd S. Lojsiewicz [1] An introduction to the theory of rel functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,, Riemnn-Stieltjes Jordn 13 5 Riemnn-Stieltjes 15 6 Riemnn-Stieltjes 22 7 Riemnn-Stieltjes 28 1

2 1 I R f : I R (incresing) ( decresing) x 1, x 2 I with x 1 < x 2 f(x 1 ) f(x 2 ) ( f(x 1 ) f(x 2 )) f(x 1 ) < f(x 2 ) (f(x 1 ) > f(x 2 )) ( (stricltly decresing)) f incresing decresing x 0 I f(x 0 + 0) = lim x x0 f(x), f(x 0 0) = lim x x0 f(x) f : I R f(x 0 + 0) = inf x 0 <x f(x), f(x 0 0) = sup x<x 0 f(x), f(x 0 + 0) = sup f(x), x 0 <x f(x 0 0) = inf x<x 0 f(x),, f : I R x 1, x 2 I with x 1 < x 2 f(x 1 0) f(x 1 ) f(x 1 + 0) f(x 2 0) f(x 2 ) f(x 2 + 0) Theorem 1.1 f (, b), f, (f(x + 0) f(x 0)) f(b 0) f( + 0) <x<b Proof., I I = [, b] f w(x) = f(x + 0) f(x 0), x f w(x) = 0. { J = {x I : w(x) > 0}, J n = x I : w(x) > 1 }, n = 1, 2,... n 2

3 , J = n=1j n. {x 1,..., x k } J n f() f(x 1 0) < f(x 1 + 0) f(x k 0) < f(x k + 0) f(b) k n w(x 1) + + w(x k ) = f(x 1 + 0) f(x 1 0) + + f(x k + 0) f(x k 0) f(b) f() k n(f(b) f()), # J n n(f(b) f()) <, J = n=1j n x 1,..., x k (, b) f(x 1 + 0) f(x 1 0) + + f(x k + 0) f(x k 0) f(mx{x 1,..., x k } 0) f(min{x 1,..., x k } + 0) f(b 0) f( + 0) Definition 1.2 {x n } N n=1, N N { } [, b], {s n } N n=1, {t n } N n=1 s n + t n > 0 n 0, if x < x n u n (x) = s n, if x = x n s n + t n, if x > x n N n=1 (s n + t n ) < 0 u n (x) s n + t n, t b u(x) = N u n (x) [, b] u sltus function. n=1 Theorem 1.3 sltus function u [, b], u {x n } n=1, u(x n ) u(x n 0) = s n, u(x n + 0) u(x n ) = t n. u [, b]\{x n } n=1. Proof. x 0 {x n } n=1 u x 0 ε > 0 N n=k+1 (s n + t n ) < 3 1 ε k. u 1,....u k x 0 δ > 0 x x 0 < δ n = 1,..., k u n (x) u n (x 0 ) < ε 3k 3

4 . x x 0 < δ k u(x) u(x 0 ) = (u n (x) u(x 0 )) + N (u n (x) u(x 0 )) n=1 n=k+1 k N u n (x) u(x 0 ) + (u n (x) u(x 0 )) u x 0 n=1 k u n (x) u(x 0 ) + n=1 < k ε 3k + 2 N n=k+1 n=k+1 N n=k+1 u u n0 = N n=1,n n 0 u n x n0 u n (x) + (s n + t n ) < ε 3 + 2ε 3 = ε N n=k+1, u(x 0 ) u(x n0 ) u(x n0 0) = (u u n0 )(x n0 ) (u u n0 )(x n0 0) + u n0 (x n0 ) u n0 )(x n0 0) = 0 + s n 0 = s n, u(x n0 + 0) u(x n0 ) = (u u n0 )(x n0 + 0) (u u n0 )(x n0 ) + u n0 (x n0 + 0) u n0 )(x n0 ) = 0 + s n + t n s n = t n Definition 1.4 [, b] f, {x n } N n=1, N N { } f, s n = f(x n ) f(x n 0), t n = f(x n + 0) f(x n ). N n=1 (s n + t n ) = N n=1 (f(x n + 0) f(x n 0)) f(b) f() sltus function u(x) = N n=1 u n(x) f sltus function. Theorem 1.5 [, b] f sltus fuction u g = f u g [, b] f(x) f(x) = u(x) + g(x), sltus function Proof. g = f u, Theorem 1.3 x n f u, x {x n } N n=1 (1.1) u(x) = {f(y + 0) f(y 0)} + f(x) f(x 0) = y<x n with x n<x {f(x n + 0) f(x n 0)} + f(x) f(x 0) 4

5 g(x) = f(x 0) y<x {f(y + 0) f(y 0)}. x 1 < x 2 b g(x 2 ) g(x 1 ) = f(x 2 0) f(x 1 0) {f(y + 0) f(y 0)} x 1 y<x 2 = f(x 2 0) f(x 1 + 0) {f(y + 0) f(y 0)} x 1 <y<x 2 Theorem 1.1. Theorem 1.6 E R f = inf E, b = sup E (, b) f E f = f Proof. f(x) = sup f (,x] E 2 E R {f n } n=1 x 1, x 2 E x 1 x 2 f n (x 1 ) f n (x 2 ) n lim n f n (x 1 ) lim n f n (x 2 ) Theorem 2.1 (Helly s First Theorem) {f n } n=1 I, x I {f n (x)} n=1 I {f n k }. Proof. I = (, b) Z (, b) I = [, b) I Z Z. ), x (, b) {f n (x)} n=1, {f n ν } ν=1 Z lim ν f nν (x), x Z Z Theorem 1.6 (, b) g(x) = lim ν f nν (x), x Z 5

6 x (, b) Z {s k }, {t k} s k < x < t k, s k, t k x, ν g(s k ) lim inf ν k g(x 0) lim inf ν f nν (s k ) f nν (x) f nν (t k ) f n ν (x) lim sup f nν (x) g(t k ) ν f n ν (x) lim sup f nν (x) g(x + 0) ν x g g(x) = lim ν f nν (x) g E E I, {f nν } ν=1 {f n νk } E {f n νk } {f n ν } ν=1 I\E, I Theorem 2.2 {f n } n=1 I = [, b], Z, b [, b] {f n } n=1 [, b] f Z, {f n } n=1 [, b] f Proof. f [, b] ε > 0 δ > 0 x x < δ f(x ) f(x) < 2 1 ε Z [, b] = x 0 < x 1 < < < x k = b x 1,... x n 1 Z x k < δ, k = 1,..., n lim n f n (x j ) = f(x j ) N N n N j = 0, 1,..., k f n (x j ) f(x j ) < 2 1 ε x [, b] x j 1 x x j j n N f n (x) f n (x j ) f(x j ) + ε ( 2 < f(x) + ε ) + ε 2 2 = f(x) + ε f n (x) f n (x j 1 ) f(x j 1 ) ε ( 2 > f(x) ε ) ε 2 2 = f(x) ε f n (x) f(x) < ε 3 Definition 3.1 [, b] f : [, b] R [, b] : = x 0 < x 1 < < x n = b (3.1) f(x k ) f( ) 6

7 , (3.2) W(f) b = sup f(x k ) f( ), f [, b] W b (f) < f. Theorem 3.2 φ : [α, β] [, b] W b (f) = W β α (f φ). Proof. [, b] : = x 0 < x 1 < < x n = b t j φ(x j ) t j, j = 1, 2,..., n 1, t 0 = α, t n = β. [α, β] : α = t 0 =< t 1 < t 2 < < t n 1 < t n = φ 1 (x n ) = β f(x k ) f( ) = f φ((t k ) f φ((t k 1 ) Wα β (f φ) W(f) b Wα β (f φ). [α, β] : α = t 0 < t 1 < < t n = β [, b] : = x 0 = φ(t 0 ) < x 1 = φ(t 1 ) < < x n 1 = φ(t n 1 ) < x n = φ 1 (t n ) = b f φ((t k ) f φ((t k 1 ) = f(x k ) f( ) W(f) b W β α (f φ) W b (f) f : [, b] R W b (f) = f(b) f(), Theorem 3.3 f : [, b] R [, b] f (x) M Lipschitz f(x) f(y) M x y W b (f) M(b ), C 1 -, W b (f) = f (x) dx Proof. Lipschitz f(x) f(y) M x y [, b] : = x 0 < x 1 < < x n = b f(x k ) f( ) M x k = M(b ) 7

8 f (x) M {ξ k } n f(x k ) f( ) x k = f (ξ k ), k = 1, 2,..., n f(x k ) f( ) = f (ξ k ) (x k ) M(x k ) = M(b ). f C 1 - [, b] : = x 0 < x 1 < < x n = b xk f(x k ) f( ) = f xk (x) dx = f (x) dx = f (x) dx W b (f) f (x) dx f (x) [, b] Riemnn ε > 0 δ > 0 [, b] : = x 0 < x 1 < < x n = b k ξ k x k {ξ k } n mesh( ) < δ f (ξ k ) (x k ) f (x) dx < ε. {ξ k } n f(x k ) f( ) x k = f (ξ k ), k = 1, 2,..., n, f (ξ k ) (x k ) = f(x k ) f( ) f(x k ) f( ) f (x) dx < ε f (x) dx ε < f(x k ) f( ) W(f) b f (x) dx W b (f) f : [, b] R, Exmple 3.4 p, q 1 < p q f : [0, 1] R { x p cos ( ) π f(x) = x, 0 < x 1 q 0, x = 0, [0, 1] W 1 0 (f) =. 8

9 Proof. x = 0 p > 1. (0, 1], ( π ) f (x) = px p 1 cos + π ( π ) x q x sin q p+1 x q p > 1 px p 1 cos ( ) ( π x π sin π ) q x q p+1 x q sin ( π x q ) 1 2, π 4 + nπ π x q 3π 4 + nπ 1 ( 3 + n) x 1 1/q ( 1 + n) 1/q π ( sin π ) dx x q p+1 x q { } π 1 1 ( ( n) 1 1/q ( 3 + n) 1/q 4 4 n=0 ( 2 n=0 n=0 ) (q p+1)/q n π 2 π ) (q p+1)/q n { (1 ) + 3 1/q ( ) } /q 4n ( n) (p q)/q ( n { ( n) 1/q ( n) 1/q } ( n) 1/q ( n) 1/q 4n ) 1/q ( ) 1/q ( ) 1/q q 1 n 1 4n 4n p q q + 1 = p q 1 Theorem 3.5 f : [, b] R W b (f) = W c (f) + W b c (f), < c < b (i) c d b W d c (f) W b (f), [, b] [c, d] (ii) c b f [, c] [c, b] [, b] 9

10 Proof. [, c] = x 0 < x 1 < < x m = c [c, b] c = y 0 < x 1 < < y n = b = x 0 < x 1 < < x m = c = y 0 < x 1 < < y n = b [, b] m f(x k ) f( ) + f(y k ) f(y k 1 ) W(f) b m f(x k ) f( ) W(f) b f(y k ) f(y k 1 ), [, c] W c W(f) b f(y k ) f(y k 1 ). f(y k ) f(y k 1 ) W(f) b W c, [c, b], W c (f) + W b c (f) W b (f). W b c W b (f) W c [, b] = x 0 < x 1 < < x n = c x j 1 c x j j c [, b] = x 0 < x 1 < < x j 1 c x j < < x n = b j 1 f(x k ) f( ) f(x k ) f( ) + f(c) f(x j 1 ) + f(x j ) f(c) + W c (f) + W b c (f) k=j+1 f(x k ) f( ) W b (f) W c (f) + W b c (f) osc [,b] f = sup f(y) f(x) x,y [,b], f [, b] (3.3) f(b) f() sup f(y) f(x) W(f) b x,y [,b] 10

11 Theorem 3.6 f : [, b] R f [, b] f(x) f() + W b (f) Theorem 3.7 [, b] f, g α, β W b (αf + βg) αw b (f) + βw b (g) [, b] Proof. [, b] = x 0 < x 1 < < x n = c αf(x k ) + βg(x k ) (αf( ) + βg( ) α f(x k ) f( + β αw b (f) + βw b (g) g(x k ) g( ) W b (αf + βg) αw b (f) + βw b (g) Theorem 3.8 E 1, E 2 R I f, g f([.b]) E 1, g([.b]) E 2. F : E 1 E 2 R Lipschitz, M > 0 F (u 1, v 1 ) F (u 0, v 0 ) M( u 1 u 0 + v 1 v 0 ), u 1, u 0 E 1 v 1, v 0 E 2 F (f, g) [, b] W b (F (f, g)) M(W b (f) + W b (g)) f, g [, b] (i) fg [, b] (ii) inf [,b] g > 0 f g [, b] Proof. [, b] = x 0 < x 1 < < x n = c F (f(x k ), g(x k )) F (f( ), g( )) { M f(x k ) f( + M(W b (f) + W b (g)) } g(x k ) g( 11

12 W b (F (f, g)) (W b (f) + W b (g)) (i) F (u, v) = uv, M = mx{sup [,b] f, sup [,b] g } E 1 = E 2 = [ M, M] F (u 1, v 1 ) F (u 0, v 0 ) = u 1 v 1 u 0 v 0 u 1 v 1 v 0 + u 1 u 0 v 0 M( u 1 u 0 + v 1 v 0 ), Theorem (ii) F (u, v) = u, A = inf v [,b] g > 0, E 1 = [ sup [,b] f, sup [,b] f ], E 2 = { } 1 (, A] [A, ) M = mx, sup [,b] f ] A A 2 F (u 1, v 1 ) F (u 0, v 0 ) = u 1 + u 0 v 1 v 0 = u 1v 0 u 0 v 1 v 1 v 0 = u 1v 0 u 0 v 0 + u 0 v 0 u 0 v 1 v 1 v 0 u 1 u 0 + u 0 v 0 v 1 M( u 1 u 0 + v 1 v 0 ) v 1 v 1 v 0 Theorem 3.9 f [, b], x = x 0 ( [, b]) [, b] x W x (f) x = x 0 ( [, b]), Proof. x > x 0 W x (f) = W x 0 (f) + Wx x 0 (f) lim x x0 +0 Wx x 0 (f) = 0 ε > 0 δ > 0 x 0 < x < x 0 + δ f(x) f(x 0 ) < 2 1 ε. [x 0, b] x 0 < x 1 < < x n = b (3.4) Wx b 0 (f) f(x 1 ) f(x 0 ) + f(x 2 ) f(x 1 ) + + f(x n ) f(x n 1 ) + ε 2 x 0 < x 1 < x 0 + δ f(x 2 ) f(x 1 ) + + f(x n ) f(x n 1 ) Wx b 1 (f) f(x 1 ) f(x 0 ) < ε 2 (3.4) Wx b 0 (f) < ε 2 + W x b 1 (f) + ε 2 = W x b 1 (f) + ε. 0 W x 1 x 0 (f) = W b x 0 (f) W b x 1 (f) < ε 12

13 Theorem 3.10 f {f n } n=1 [, b], x [, b] lim n f n (x) W b (f) lim inf n W b (f n ) Proof. lim inf n W b (f n ) < ε > 0 {f nν } ν=1 W b (f nν ) < lim inf n W b (f n ) + ε = x 0 < x 1 < < x n = b ν f nν (x k ) f nν ( ) W b (f nν ) < lim inf n f(x k ) f( ) lim inf n W b (f n ) + ε W b (f n ) + ε W b (f) lim inf n W b (f n ) + ε, ε +0 W b (f) lim inf n W b (f n ) 4 Jordn [, b], 2 Theorem 3.7 Theorem 4.1 (Jordn ) f [, b] (4.1) φ(x) = 1 2 {W x (f) + f(x)}, ψ(x) = 1 2 {W x (f) f(x)}, [, b], f x 0 ( [, b]) ( ) φ, ψ x 0 ( ) (i) f = φ ψ 2 (ii) φ ψ f f = φ ψ f 2 x 0 < x 1 b φ(x 1 ) φ(x 0 ) φ(x 1 ) φ(x 0 ), ψ(x 1 ) ψ(x 0 ) ψ(x 1 ) ψ(x 0 ) 13

14 (iii) W x (f) = W x (φ) + W x (ψ) x [, b] Proof. x 0, x 1 [, b] with x 0 < x 1 b f(x 1 ) f(x 0 ) W x 1 x 0 φ(x 1 ) φ(x 0 ) = 1 2 {W x 1 (f) + f(x 1 )} 1 2 {W x 0 (f) + f(x 0 )} = 1 2 { f(x1 ) f(x 0 ) + W x 1 x 0 } 0, ψ(x 1 ) ψ(x 0 ) = 1 2 {W x 1 (f) f(x 1 )} 1 2 {W x 0 (f) f(x 0 )} = 1 2 { f(x1 ) + f(x 0 ) + W x 1 x 0 } 0, φ, ψ Theorem 3.9. (i) (ii) φ(x 1 ) φ(x 0 ) φ(x 1 ) φ(x 0 ) 1 2 {W x 1 (f) + f(x 1 )} 1 2 {W x 0 (f) + f(x 0 )} φ(x 1 ) φ(x 0 ) 1 2 { W x 1 x 0 (f) + f(x 1 ) f(x 0 ) } φ(x 1 ) φ(x 0 ) W x 1 x 0 (f) + φ(x 1 ) φ(x 0 ) ( ψ(x 1 ) ψ(x 0 )) 2( φ(x 1 ) φ(x 0 )) W x 1 x 0 (f) φ(x 1 ) φ(x 0 ) + ψ(x 1 ) ψ(x 0 ) ψ(x 1 ) ψ(x 0 ) ψ(x 1 ) ψ(x 0 ) 1 2 {W x 1 (f) f(x 1 )} 1 2 {W x 0 (f) f(x 0 )} ψ(x 1 ) ψ(x 0 ) 1 2 { W x 1 x 0 (f) f(x 1 ) + f(x 0 ) } ψ(x 1 ) ψ(x 0 ) W x 1 x 0 (f) φ(x 1 ) + φ(x 0 ) + ψ(x 1 ) ψ(x 0 ) 2( ψ(x 1 ) ψ(x 0 )) W x 1 x 0 (f) φ(x 1 ) φ(x 0 ) + ψ(x 1 ) ψ(x 0 ), x 0 = y 0 < y 1 < < y n = x 1 f(y k ) f(y k 1 ) = φ(y k ) φ(y k 1 ) ψ(y k ) ψ(y k 1 ) ( φ(y k ) φ(y k 1 )) + [x 0, x 1 ] ( ψ(y k ) ψ(y k 1 )) = φ(x 1 ) φ(x 0 ) + ψ(x 1 ) ψ(x 0 ) 14

15 (iii) φ ψ (4.1) φ(x) + ψ(x) = W x (f), φ, ψ W x (φ) = φ(x) φ(), W x (ψ) = ψ(x) ψ(), φ() = 1 2 {W (f) + f()} = f() 2, ψ() = 1 2 {W (f) f()} = f() 2, Jordn, Theorem 4.2 () f [, b] x 0 f(x 0), f(x + 0), f( + 0), f(b 0) (b) f, (c) f [, b] f = g 1 + u 1 (g 2 + u 2 ) g 1, g 2, sltus function u 1, u 2 (d) (Helly s First Theorem) {f n } n=1 [, b], {f n()} n=1 {W b (f n )} n=1, [, b] {f n ν } ν=1 (d) lim ν f nν (x) Theorem Riemnn-Stieltjes f, g [, b] [, b] : = x 0 < x 1 < < x n = b ξ k x k, k = 1, 2,..., n {ξ k } S(, {ξ k }) = f(ξ k )(g(x k ) g( )). mesh( ) = mx{x k : k = 1, 2,..., n}. f g Riemnn-Stieltjes, l ε > 0 : δ > 0 : nd {ξ k } with mesh( ) < δ : S(, {ξ k }) l < ε 15

16 . f(x) dg(x) = l φ : [α, β] [, b] f g Riemnn-Stieltjes f φ g φ Riemnn-Stieltjes, f(x) dg(x) = β f(φ(t))dg(φ(t)) α Riemnn-Stieltjes dg(x) = g(b) g(), g f(x) dx = 0 c (, b) { 0, x < c H c (x) = 1, c x b, f c f(x) dh c (x) = f(c) Theorem 5.1 [, b] f, g C 1 - f(x) dg(x) = f(x)g (x) dx Proof. = x 0 < x 1 < < x n = b {ξ k } S(, {ξ k }) = f(ξ k )(g(x k ) g( )) = f(ξ k )g (ξ k){x k } f(x)g (x) dx S(, {ξ k }) xk = {f(x)g (x) f(ξ k )g (ξ k)} dx xk f(x)g (x) f(ξ k )g (ξ k) dx xk { f(x) g (x) g (ξ k) + f(x) f(ξ k ) g (ξ k) } dx {mx [,b] f osc [xk 1,x k ] g + mx g osc [xk 1,x k ] f}(b ) [,b] f, g, mesh( ) 0 0. f(x) dg(x) = f(x)g (x) dx 16

17 Definition 5.2. Lemm 5.3 Riemnn-Stieltjes f(x) dg(x) = l ε > 0 δ > 0 mesh( ) < δ, {x k }, {x k } S(, {ξ k }) S(, {ξ k}) < ε Riemnn-Stieltjes, Riemnn-Stieltjes mesh( 1 ) < δ, mesh( 2 ) < δ 2 1, 2 S( 1, {ξ k }) S( 2, {ξ k }) S( 1, {ξ k }) S(, {ξ k}) + S( 2, {ξ k }) S(, {ξ k}) < 2ε, Cychy Lemm 5.4 : = x 0 < x 1 < < x n = b, S(, {ξ k }) S(, {ξ k}) ( osc[xk 1,x k ] f ) W x k (g) Proof. : = x 0 < x 1 < < x m = b x α k = x k α k S(, {ξ k}) S(, {ξ k }) α l = f(ξ ν)(g(x ν) g(x ν 1)) f(ξ k ) ν=α k 1 +1 α l = (f(ξ k)) f(ξ ν))(g(x ν) g(x ν 1)) ν=α k 1 +1 α l osc [xk 1,x k ] f g(x ν) g(x ν 1) ν=α k 1 +1 osc [xk 1,x k ] f W x k (g). α l ν=α k 1 +1 (g(x ν) g(x ν 1)) Theorem 5.5 f g Riemnn-Stieltjes f(t) dg(t) 17

18 Proof. f ε > 0 δ > 0 x y δ f(x) f(y) < ε/(w b (g)+1) mesh( ) < δ osc [xk 1,x k ] fε/(w b (g)+1) Lemm S(, {ξ k}) S(, {ξ k }) osc [xk 1,x k ] f W x k (g). ε W(g) b + 1 W x W b k (g) = ε (g) W(g) b + 1 ε Lemm 5.3 Riemnn-Stieltjes f(t) dg(t) Lemm 5.6 Riemnn-Stieltjes f(x) dg(x), : = x 0 < x 1 < < x n = b osc [xk 1,x k ] f < ε, k = 1, 2,..., n S(, {ξ k}) f(x) dg(x) εw (g) b Proof. η > 0 δ > 0 mesh( ) < δ f(x) dg(x) S(, {ξ k}) < η S(, {ξ k }) S(, {ξ k}) < εw(g) b S(, {ξ k}) f(x) dg(x) S(, {ξ k }) S(, {ξ k}) + <εw b (g) + η f(x) dg(x) S(, {ξ k}) η > 0, Theorem 5.7 [, b] f, g Riemnn-Stieltjes c [, b] f g Proof. ε > 0 Lemm 5.3 δ > 0. mesh( ) < δ S(, {ξ k}) S(, {ξ k}) < ε 18

19 {f(ξ k) f(ξ k)}{g(x k ) g( )} < ε. g(x k ) g( ) ξ k ξ k {f(ξ k ) f(ξ k )}{g(x k) g( )} 0 {ξ k )}, {ξ k )} osc [xk 1,x k ] f g(x k ) g( ε g c [, b] f c c < b g(c + 0) g(c) g(x 0 + 0) g(x 0 0) 0 < h < min{δ, b c} h x h = c, x h = c h, mesh( ) < δ, x h c + h osc [xh,c+h] f g(c + h) g(x h ) ε lim h +0 g(c + h) g(x h ) > 0, f c Theorem 5.8 f [, b] g [, b], 3 (A) Riemnn-Stieltjes f(t) dg(t) (B) ε > 0 δ > 0 mesh( ) < δ osc [xk 1,x k ] f g(x k ) g( ) < ε (C) ε > 0 δ > 0 mesh( ) < δ osc [xk 1,x k ] f W x k (g) < ε Proof. Theorem 5.7 (A) = (B) Lemm (C) = (A) (B) = (C) (B) M = osc [,b] f. M = 0 (C), M > 0 ε > 0 δ 0 > 0 mesh( ) < δ 1 osc [xk 1,x k ] f (g(x k ) g( )) < ε 3 19

20 W(g) b ε N 3M g(z i ) g(z i 1 ) i=1 = z 0 < z 1 < < z N = b, δ 2 = min{z 1 z 0,..., z N z N 1 } > 0. Theorem 5.7 (B) [, b] c f g g c Theorem 3.9, x W x (g) x = c, f W x (g) f, W x (g) δ 3 > 0 i = 1, 2,..., N x < z i < x, x x < δ 1 osc [x,x] f Wx x (g) < ε 3N mesh( ) < δ 0 = min{δ 1, δ 2, δ 2 } : = x 0 < < x n = x 0 < < x m {z 1,..., z N 1 } = z 0,..., z N = b W(g) b ε N 3M g(z i ) g(z i 1 ) m m S 0 = osc [x k 1,x k ] f W x k (g) m M i=1 {W x k x k 1 m g(x k) g(x k 1) g(x k) g(x k 1) } osc [x k 1,x k ] f {W x k (g) g(x k) g(x k 1) } + m {W x k (g) g(x k) g(x k 1) } + ε 3 < 2ε 3 S = ε 3M m osc [x k 1,x k ] f g(x k) g(x k 1) osc [xk 1,x k ] f W x k (g), < z i < x k k S, S, S N ε 3N = ε 3, S 0, S S 0 < 2ε 3, S S + S < ε (C) 20

21 Theorem 5.9 g = φ ψ [, b] g Jordn [, b] f g Riemnn-Stieltjes φ ψ Riemnn-Stieltjes, f(x) dg(x) = f(x) dφ(x) f(x) dψ(x) Proof. Theorem 4.1 Jordn W x k (g) = W x k (φ) + W x k (ψ) Theorem 5.8 (C) Theorem 5.10 f [, b], g, f g Riemnn- Stieltjes, [c, d] [, b] f g Riemnn-Stieltjes Proof. [c, d] mesh( ) [, b], Theorem 5.8 (C). Remrk 5.11 f g [, c] [c, b] Riemnn-Stieltjes, [, b] Riemnn-Stieltjes f c jump, g c jump,. Theorem 5.12 f, g, [, b] Riemnn-Stieltjes f(x) dg(x), f, g Proof., Theorem 5.7., ε > 0 ε ε 1 = W(f) b + W(g) b. (W(f) b = W(g) b = 0 f, g Riemnn-Stieltjes f(x) dg(x), W b (f) + W(g) b > 0 ). δ > 0 x, x x x < δ osc [x,x ] f < ε 1 g(x ) g(x ) < ε 1, δ > 0 x x < δ osc [x,x ] f ε 1 g(x ) g(x ) ε 1 x, x [, b] n δ = 1 n, x, x, x n, x n, x n x n < 1 n osc [x n,x n] f ε 1 g(x n) g(x n) ε 1 21

22 ξ n, x n f(ξ n) f(ξ n) ε 1 2 x n k x 0, x n k, ξ n k, ξ n k x 0, f, g x 0, δ > 0 : = x 0 < < x n mesh( ) < δ osc [xk 1,x k ] f < ε 1 g(x k ) g( ) < ε 1 fork = 1,..., n osc [xk 1,x k ] f g(x k ) g( ) ε 1 g(x k ) g( ) + sum n ε 1 osc [xk 1,x k ] f εw b (g) W(f) b + W(g) + εw(f) b b W(f) b + W(g) = ε b, f g Riemnn-Stieltjes 6 Riemnn-Stieltjes Theorem 6.1 [, b] f [, b] g Riemnn-Stieltjes b f(x) dg(x) sup f W(g). b [,b] Proof. f(x) dg(x) f(ξ k ){g(x k ) g( )} f(ξ k ) g(x k ) g( ) sup f [,b] g(x k ) g( ) sup f W(g) b [,b] Theorem 6.2 [, b] f 1, f 2 [, b] g Riemnn- Stieltjes f 1 (x) f 2 (x) [, b] f 1 (x) dg(x) f 1 (x) dg(x). 22

23 Proof., g(x k ) g( ) 0 f 1 (ξ k ){g(x k ) g( )} f 2 (ξ k ){g(x k ) g( )} Theorem 6.3 [, b] f [, b] g Riemnn-Stieltjes, c (, b) f [, c] [c, b] g Riemnn-Stieltjes f(x) dg(x) = c f(x) dg(x) + c f(x) dg(x). Proof. Theorem 5.10 : = x 0 < < x k = c = y 0 < < y l = b [, b] mesh( ) 0 [, c], [c, b] 1 : = x 0 < < x k = c, 2 : c = y 0 < < y l = b mesh( 1 ) 0, mesh( 2 ) 0, k f(ξ i ){g(x i ) g(x i 1 )} + i=1, f(x) dg(x) c f(x) dg(x) + c l f(η i ){g(y i ) g(y i 1 )} i=1 f(x) dg(x), Theorem 6.4 [, b] f 1, f 2 [, b] g Riemnn- Stieltjes, c 1, c 2 c 1 f 1 + c 2 f 2 g Riemnn- Stieltjes {c 1 f 1 (x) + c 2 f 2 (x)} dg(x) = c 1 f 1 (x) dg(x) + c 2 f 2 (x) dg(x). Proof. x, y [, x k ] c 1 f 1 (y)+c 2 f 2 (y) (c 1 f 1 (x)+c 2 f 2 (x)) c 1 f 1 (y) f 1 (x) + c 2 f 2 (y) f 2 (x) osc [xk 1,x k ] f 1 +osc [xk 1,x k ] f 2 osc [xk 1,x k ](c 1 f 1 +c 2 f 2 )W x k (g) c 1 osc [xk 1,x k ](c 1 f 1 + c 2 f 2 ) osc [xk 1,x k ] f 1 + osc [xk 1,x k ] f 2 osc [xk 1,x k ] f 1 W x k (g)+c 2 osc [xk 1,x k ] f 2 W x k (g) Theorem 5.8, c 1 f 1 + c 2 f 2 g Riemnn- Stieltjes, {c 1 f 1 (ξ k )+c 2 f 2 (ξ k )}{g(x k ) g( )} = c 1 f 1 (ξ k ){g(x k ) g( )}+c 2 f 2 (ξ k ){g(x k ) g( )} mesh( ) 0 23

24 Theorem 6.5 [, b] f [, b] g 1, g 2 Riemnn-Stieltjes, c 1, c 2 f c 1 g 1 + c 2 g 2 Riemnn-Stieltjes f(x) d{c 1 g 1 (x) + c 2 g 2 (x)} = c 1 f(x) dg 1 (x) + c 2 f(x) dg 2 (x). Proof. W x k(c 1 g 1 +c 2 g 2 ) c 1 W x k(g 1 + c c W x k(g 2 Theorem 5.8, f c 1 g 1 + c 2 g 2 Riemnn-Stieltjes, f(ξ k ){c 1 g 1 (x k ) + c 2 g 2 (x k ) (c 1 g( ) + c 2 g 2 ( )} =c 1 f(ξ k ){g 1 (x k ) g 1 ( )} + c 2 f(ξ k ){g 2 (x k ) g 2 ( )} mesh( ) 0 Theorem 6.6 [, b] f [, b] g Riemnn-Stieltjes, µ [inf [,b] f, sup [,b] f] f(x) dg(x) = µ{g(b) g()} Proof. g g(b) = g() f(x) dg(x) = 0, µ R g(b) > g(), Theorem 6.2 µ = f(x) dg(x) g(b) g() inf f {g(b) g()} = [,b] inf [,b] f dg(x) f(x) dg(x) sup [,b] f dg(x) = sup f {g(b) g()} [,b] g(b) g() µ [inf [,b] f, sup [,b] f]. Theorem 6.7 ( ) f, g [, b], f(x) dg(x) + 24 f(x) dg(x) = f(b)g(b) f()g()

25 Proof. f g, g f Riemnn-Stieltjes Theorem : = x 0 < < x n = b f(x k ){g(x k ) g( )} + g( ){f(x k ) f( )} = f(b)g(b) f()g(), mesh( ) 0 Theorem 6.8 ( ) [, b] φ, g Riemnn- Stieltjes G(x) = x φ(x) dg(x) [, b] f f G Riemnn-Stieltjes fφ g Riemnn-Stieltjes, f(x) dg(x) = f(x)φ(x) dg(x) Proof. [c, d] [, b] d G(d) G(c) = φ(x) dg(x) sup φ Wc d (g) sup φ Wc d (g) [c,d] [,b] c G(x k ) G( ) sup φ W x k (g) sup [,x k ] [,b] (g) = sup W(g) b W x k [,b] G : = x 0 < < x n = b {ξ k } f(ξ k )(G(x k ) G( )) f(ξ k )φ(ξ k )(g(x k ) g( ) xk xk = f(ξ k ) φ(x) dg(x) f(ξ k )φ(ξ k ) dg(x) xk = f(ξ k ){φ(x) φ(ξ k )} dg(x) sup [,x k ] f osc [xk 1,x k ] φ W x k (g) sup f osc [xk 1,x k ] φ W x k (g) [,b] φ g [, b] Riemnn-Stieltjes mesh( ) 0 rightrrow0. f G Riemnn-Stieltjes, 25

26 n f(ξ k)(g(x k ) G( )) f(x) dg(x) lim n mesh( ) 0 f(ξ k)φ(ξ k )(g(x k ) g( ) f(x)φ(x) dg(x), fφ g Riemnn-Stieltjes f(x) dg(x) = f(x)φ(x) dg(x) fφ g Riemnn-Stieltjes Theorem 6.9 n N, f n [, b], g Riemnn-Stieltjes {f n } n=1, f [, b], f g Riemnn-Stieltjes Proof. lim n f n (x) dg(x) = f(x) dg(x) ε n = sup [,b] f n (x) f(x), ε n 0. [c, d] [, b] f n (x) f n (y) f n (x) f(x) + f(x) f(y) + f(y) f n (y) 2ε n + osc [c,d] f, x, y [c, d] osc [c,d] f n 2ε n + osc [c,d] f f n f, osc [c,d] f 2ε n + osc [c,d] f n osc [c,d] f n osc [c,d] f 2ε n ε > 0 n 0 N ε n0 < ε 4(W b (g) + 1), δ > 0 mesh( ) < δ N osc [xk 1,x k ] f n0 W x k (g) < ε 2 N osc [xk 1,x k ] f W x k (g) N N ε 2 + (osc [xk 1,x k ] f n0 + 2ε n0 ) W x k (g) osc [xk 1,x k ] f n0 W x k (g) + 2ε n0 W b (g) 2ε 4(W b (g) + 1) W b (g) < ε 26

27 f g Riemnn-Stieltjes Theorem 6.4 Theorem 6.1 b f n (x) dg(x) f n (x) dg(x) = (f n (x) f(x)) dg(x) ε nm(g) b 0 (n ) Theorem 6.10 (Helly s Second Theorem) f [, b] g {g n } n=1 [, b],, b Z [, b] Z lim n g n (x) = g(x) {W(g b n )} n=1 lim n f(x) dg n (x) = f(x) dg(x) Proof. φ n = g n g lim n f(x) dφ n (x) = 0 W b (φ n ) W b (g n ) + W b (g) W b (φ n ) M, n = 1, 2,... M. ε > 0 f δ > 0 mesh( ) < δ osc [xk 1,x k ] f ε 2M Lemm 5.6 N f(x) dφ n (x) f(ξ k )(φ n (x k ) φ n ( )) ε 2M W b (φ n ) ε 2 : = x 0 < c < x n = b Z, n 0 N n n 0 N f(ξ k )(φ n (x k ) φ n ( )) < ε 2 2 f(x) dg(x) < ε. lim n f(x) dg n (x) = f(x) dg(x) 27

28 7 Riemnn-Stieltjes µ R Borel B(R) ( µ(r) < ) C c (R), R support Theorem 7.1 g(x) = µ((, x]), x R, lim x g(x) = 0, lim x g(x) = µ(r). f C c (R) f(x)dµ(x) = f(x) dg(x) R supp f [, b] [, b] f(x) dg(x) f(x) dg(x) = Proof. f(x) dg(x) [, b] [, b] [A, B] f(x) dg(x) = B f(x) dg(x) Theorem 6.3 A [A, ] [b, B] f 0 B A f(x) dg(x) = A f(x) dg(x) + f(x) dg(x) + B b f(x) dg(x) = f(x) dg(x) ε > 0 Riemnn-Stieltjes Theorem 5.8 δ > 0 mesh( ) < δ : = x 0 < < x n = b, {ξ k } n f(x) dg(x) f(ξ k )(g(x k ) g( )) < ε 2 osc [xk 1,x k ] f g(x k ) g(f(ξ k )) < ε 2 f(x)dµ(x) = f(x)dµ(x) = R (,b] f(x)dµ(x) (,x k ] f(ξ k )(g(x k ) g( )) = f(ξ k )µ((, x k ]) = f(ξ k )dµ(x) (,x k ] 28

29 f(x) dµ(x) f(x) dg(x) R f(x) dµ(x) f(ξ k )(g(x k ) g( )) R + f(x) dg(x) = (f(x) f(ξ k ))dµ(x) (,x k ] + ε 2 osc [xk 1,x k ] f dµ(x) + ε (,x k ] 2 = osc [xk 1,x k ] f g(x k ) g( ) + ε 2 < ε 2 + ε 2 = ε. f(ξ k )(g(x k ) g( )) Remrk 7.2 f, supp f [, b] [, b] f g Riemnn-Stieltjes f(x)dµ(x) = f(x) dg(x) R µ R, [, b] Borel B([, b]) g(x) = µ([, x]), x b f(x)dµ(x) = f(x) dg(x) [,b] Exmple 7.3 µ unit mss Dirc, f [,b] f(x) dµ(x) = f(), g(x) 1, f(x) dg(x) = 0. g Theorem 7.4 g(x) = { 0, x = µ([, b]), < x b, (, b], g() = 0, g(b) = µ([, b]). f C([, b]) f(x)dµ(x) = f(x) dg(x) [,b] 29

30 Proof. ε > 0 Riemnn-Stieltjes Theorem 5.8 δ > 0 mesh( ) < δ : = x 0 < < x n = b, {ξ k } n f(x) dg(x) f(ξ k )(g(x k ) g( )) < ε 3 osc [xk 1,x k ] f g(x k ) g(f(ξ k )) < ε 3 δ > 0 f f(x) f() µ({}) < ε 3, x x 1 [,b] f(x)dµ(x) = f()µ({}) + (,b] f(x)dµ(x) = f()µ({}) + (,x k ] f(x)dµ(x) g(x 1 ) g(x 0 ) = g(x 1 ) g() = g(x 1 ) = µ([, x 1 ]) = µ({}) + µ((, x 1 ]) f(ξ k )(g(x k ) g( )) =f(ξ 1 )µ({}) + f(ξ k )µ((, x k ]) =f(ξ 1 )µ({}) + f(x) dµ(x) f(x) dg(x) [,b] f(x) dµ(x) f(ξ k )(g(x k ) g( )) [,b] + f(x) dg(x) = (f() f(ξ 1))µ({}) + (f(x) f(ξ k ))dµ(x) (,x k ] + ε 3 f() f(ξ 1 ) µ({}) + osc [xk 1,x k ] f dµ(x) + ε (,x k ] 3 = ε 3 + osc [xk 1,x k ] f g(x k ) g( ) + ε 3 < ε 3 + ε 3 + ε 3 = ε. (,x k ] f(ξ k )dµ(x) f(ξ k )(g(x k ) g( )) 30

31 [, b] g g(), x = g(), x = g + (x) = lim x x+0 g(x), < x < b, g (x) = lim x x 0 g(x), < x < b g(b), x = b g(b), x = b x 1 < x 2 b g() g (x 1 ) g(x 1 ) g + (x 1 ) g (x 2 ) g(x 2 ) g + (x 2 ) g(b), g, g + (, b) g ± g, g ± () = g(), g ± (b) = g(b) Theorem 7.5 ( Riemnn-Stieltjes ) [, b] f [, b] g Riemnn-Stieltjes f g ± Riemnn-Stieltjes, f(x) dg ± (x) = f(x) dg(x) Proof., g + ε > 0 δ > 0 mesh( ) < δ : = x 0 < < x n = b {ξ k } n f(x) dg(x) f(ξ k )(g(x k ) g( )) < ε 3 osc [xk 1,x k ] f W x k (g) < ε 3 31

32 mesh( ) < δ 2 {ξ k} f(ξ k )(g + (x k ) g + ( )) f(ξ k )(g(x k ) g( )) = f(ξ k )(g + (x k ) g(x k )) f(ξ k )(g + ( ) g( )) = f(ξ n 1 n)(g + (b) g(b)) f(ξ 1 )(g + () g()) + (f(ξ k ) f(ξ k+1 ))(g + (x k ) g(x k )) n 1 = (f(ξ k ) f(ξ k+1 ))(g + (x k ) g(x k )) n 1 f(ξ k ) f(ξ k+1 ) g + (x k ) g(x k ) k ξ k x k ξ k+1 x k+1 x 0, x 2,... b [, b], x 1, x 3,... b [, b] : = y 0 < < y k = b, : = z 0 < < z l = b,. mesh( ) 2 mesh( ) < δ mesh( ) 2 mesh( ) < δ n 1 f(ξ k ) f(ξ k+1 ) (g + (x k ) g(x k ) k i=1 osc [yi 1,y i ] f W y i y i 1 (g) + l i=1 osc [zi 1,z i ] f W z i z i 1 (g) < ε 3 + ε 3 = 2ε 3 mesh( ) < δ 2 {ξ k} f(x) dg(x) f(ξ k )(g + (x k ) g + ( )) f(x) dg(x) f(ξ k )(g(x k ) g( )) + f(ξ k )(g + (x k ) g + ( )) f(ξ k )(g(x k ) g( )) ε 3 + 2ε 3 = ε. f g + Riemnn-Stieltjes f(x) dg+ (x) = f(x) dg(x) 32

33 [1] S. Lojsiewicz, An introduction to the theory of rel functions, John Wiley & Sons, Ltd., Chichester,

34 Helly s first theorem, 15 Helly s second theorem, 27 Jordn, 13 Lipschitz, 11 Riemnn-Stieltjes, 15 sltus function, 3 (stricltly decresing), 2 (strictly incresing), 2 (decresing), 2 (oscilltion), 10 (incresing), 2 (monotone), 2 (integrtion by prts), 24 (subdivision), 6 (function of bounded vrition), 7 34

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

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