Properties of the Riemann Stieltjes Integral


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1 Properties of the Riemnn Stieltjes Integrl Theorem (Linerity Properties) Let < c < d < b nd A,B IR nd f,g,α,β : [,b] IR. () If f,g R(α) on [,b], then Af +Bg R(α) on [,b] nd [ ] b Af +Bg dα A +B (b) If f R(α) R(β) on [,b], then f R(Aα+Bβ) on [,b] nd f d(aα+bβ) A +B (c) If f R(α) on [,c] nd on [c,b], then f R(α) on [,b] nd c + (d) If f R(α) on [,b] then f R(α) on [c,d] [,b]. c f dβ Proof: () Let ε > 0. Then, for ny prtition IP of [,b] nd choice T for IP, S(IP,T,Af +Bg,α) A B AS(IP,T,f,α)+BS(IP,T,g,α) A B A S(IP,T,f,α) + B S(IP,T,g,α) Assume tht A nd B re nonzero. (The cses tht A nd/or B re zero re similr, but esier.) Since f R(α) on [,b] there is prtition IP f,ε such tht S(IP,T,f,α) < ε whenever IP IP f,ε 2 A nd since g R(α) on [,b] there is prtition IP g,ε such tht S(IP,T,g,α) < ε whenever IP IP g,ε 2 B It now suffices to set IP ε IP f,ε IP g,ε nd observe tht S(IP,T,Af +Bg,α) A B < ε (b) See Problem Set 1, #3. (c) See Problem Set 1, #2. (d) See Problem Set 2, #3. whenever IP IP ε c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 1
2 Theorem (Integrtion by Prts) Let < b nd f,α : [,b] IR. If f R(α) on [,b], then α R(f) on [,b] nd f(x)dα(x)+ α(x)df(x) f(b)α(b) f()α() Remrk. () The integrtion by prts formul my lso be written + αdf d(fα) (b) We shll lter see tht if α hs continuous first derivtive, then f(x)dα(x) f(x)α (x)dx. So if both f nd α hve continuous first derivtives, we my write the integrtion by prts formul s f(x)α (x)dx f(b)α(b) f()α() α(x)f (x)dx which is the integrtion by prts formul of first yer clculus (though you probbly used f(x) u(x) nd α(x) v(x). Proof of integrtion by prts: Our gol is to show tht αdf exists nd tkes the vlue αdf f(b)α(b) f()α() So let s look t the difference between Riemnn sum for αdf nd the right hnd side. For ny prtition IP { x 0, x 1, x 2, x 3,, x n b of [,b] nd choice T { t 1, t 2, t 3,, t n for IP, Note tht S ( IP,T,α,f ) { f(b)α(b) f()α() α(t i ) [ ] n f(x i ) f(x i 1 α(x i )f(x i ) + {{ α(b)f(b) for in α(x i 1 )f(x i 1 ) + {{ α()f() for (The 1 i n 1 terms of the second sum cncel the 2 i n terms of the third sum.) f(x i ) [ α(x i ) α(t i ) ] f(x j 1 ) [ α(t j ) α(x j 1 ) ] + j1 c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 2
3 f(x i ) [ α(x i ) α(t i ) ] looks like term in Riemnn sum pproximtion to with subintervl [t i,x i ] nd choice point x i [t i,x i ] nd f(x j 1 ) [ α(t j ) α(x j 1 ) ] looks like term in Riemnn sum pproximtion to with subintervl [x j 1,t j ] nd choice point x j 1 [x j 1,t j ]. Here is figure which shows ll of these subintervls. x 0 j 1 i 1 j 2 i i n 1 j i n t 1 x 1 t 2 2x 2 t n 1 x n 1 nt n x n b So ll of the subintervls fit together perfectly to form the prtition 1 nd where the choice IP T { x 0, t 1, x 1, t 2, x 2, t 3,, x n 1, t n, x n S ( IP,T,α,f ) T { j1 x 0, x 1, { f(b)α(b) f()α() j2 x 1, i2 x 2, S ( IP T,T,f,α ) + j3 x 2, i3 in 1 x 3,, x n 1, jn x n 1, Now let ε > 0. Since f R(α) for [,b] there is prtition IP ε such tht S( ) b IP, T,f,α < ε in x n for ll prtitions IP finer thn IP ε. If the prtitition IP bove is finer thn IP ε then the prtition IP T is lso finer thn IP ε nd we hve S( IP,T,α,f ) { f(b)α(b) f()α() S( IP T,T,f,α ) < ε Theorem (The Chnge of Vribles x g(y)) Let < b nd c < d, g : [c,d] [,b] be continuous, strictly monotoniclly incresing, nd obey g(c) nd g(d) b nd f,α : [,b] IR. Set h(y) f ( g(y) ) β(y) α ( g(y) ) If f R(α) on [,b], then h R(β) on [c,d] nd d c h(y)dβ(y) f(x)dα(x) 1 There is subtlety hidden in these definitions. We re not llowed to use subintervls of width zero. So if, for exmple, t i x i, we merge the two subintervls [x i 1,t i ], [t i,x i ] into the single subintervl [x i 1,x i ] nd use f(x i 1 )[α(t i ) α(x i 1 )]+f(x i )[α(x i ) α(t i )] f(x i 1 )[α(x i ) α(x i 1 )]. c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 3
4 Proof: Our gol is to prove tht d c h(y)dβ(y) exists nd equls f(x)dα(x), so let s consider the difference between Riemnn sum for d c h(y)dβ(y) nd f(x)dα(x). For ny prtition IP {c y 0,,y n d of [c,d] nd ny choice T {s 1,,s n for IP S(IP,T,h,β) n h(s i ) [ β(y i ) β(y i 1 ) ] f ( g(s i ) )[ α ( g(y i ) ) α ( g(y i 1 ) )] S(g(IP),g(T),f,α) where g(ip) { g(y) y IP { x0 g(y 0 ) g(c), x 1 g(y 1 ),, x n g(y n ) g(d) b is prtition of [,b] becuse g is ssumed to be strictly monotonic, so tht y i 1 < y i x i 1 < x i nd is ssumed to obey x 0 g(y 0 ) nd x n g(y n ) b nd g(t) { t1 g(s 1 ),, t n g(s n ) is choice for g(ip) becuse g is ssumed to be strictly monotonic so tht y i 1 s i y i x i 1 g(y i 1 ) g(s i ) t i g(y i ) x i Now let ε > 0. We hve ssumed tht f R(α) on [,b]. So there is prtition IP f,ε of [,b] such tht IP f IP f,ε S(IP f,t f,f,α) < ε for ll choices T f for IP f. Thessumptionsthtwehvemdeong gurnteethttheinversefunctiong 1 : [,b] [c,d] exists nd tht g 1( IP f,ε ) is prtition of [c,d]. We choose IPε g 1( IP f,ε ). Then s desired. IP IP ε g(ip) g(ip ε ) IP f,ε S(IP,T,h,β) S( finer thn IP f,ε g(ip), llowed T f g(t),f,α) < ε c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 4
5 Theorem (Second Fundmentl Theorem of Clculus) Let < b nd f : [, b] IR. Assume tht f is differentible on [,b] nd f R on [,b] Then f (x)dx f(b) f() Proof: Let IP { x 0,x 1,x 2,,x n be ny prtition of [,b]. Then, by the men vlue theorem, there exists, for ech 1 i n t i [x i 1,x i ] with So, setting T { t 1,t 2,,t n, we hve S(IP,T,f ) f(x i ) f(x i 1 ) f (t i ) ( x i x i 1 ) f (t i ) ( ) n [ x i x i 1 f(xi ) f(x i 1 ) ] f(b) f() So now we just hve to pply the definition of f R on [,b]. Theorem (Bsic Bounds) Let < b nd f,g,α : [,b] IR. Assume tht f,g R(α) on [,b] nd α is monotoniclly incresing. () If f(x) g(x) for ll x [,b], then () If f(x) g(x) for ll x [,b], then Proof: Let ε > 0. Since f R(α) on [,b] there is prtition IP f,ε such tht S(IP,T,f,α) < ε whenever IP IP f,ε nd T is choice for IP. Since g R(α) on [,b] there is prtition IP g,ε such tht b S(IP,T,g,α) < ε whenever IP IP g,ε nd T is choice for IP. Set IP ε { x 0,x 1,x 2,,x n b IP f,ε IP g,ε nd let T { t 1,t 2,,t n be choice for IP ε. c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 5
6 () We hve S(IP ε,t,f,α)+ε f(t i ) [ α(x i ) α(x i 1 ) ] +ε g(t i ) [ α(x i ) α(x i 1 ) ] +ε (since f(t i ) g(t i ) nd α(x i ) α(x i 1 ) 0) S(IP ε,t,g,α)+ε +2ε As +2ε is true for ll ε > 0, we lso hve. (b) We hve As S(IPε,T,f,α) +ε f(ti ) [ α(x i ) α(x i 1 ) ] +ε g(t i ) [ α(x i ) α(x i 1 ) ] +ε (since f(t i ) g(t i ) nd α(x i ) α(x i 1 ) 0) S(IP ε,t,g,α)+ε +2ε +2ε is true for ll ε > 0, we lso hve. Theorem (First Fundmentl Theorem of Clculus) Let < b nd f : [,b] IR. Assume tht f R on [,b] Set, for x b, Then () F is continuous on [, b] nd F(x) x f(x)dx (b) if f is continuous t x 0 [,b], then F is differentible t x 0 nd F (x 0 ) f(x 0 ). c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 6
7 Proof: () Since f R it is bounded. Suppose tht f(t) M for ll t b. Then F(y) F(x) y x f(t)dt M y x so F is uniformly continous. (b) Let f be continuous t x 0 [,b]. Then, for ll x b F(x) F(x 0 ) x f(x 0 ) x x 0 x 0 f(t)dt f(x 0 )[x x 0 ] x x 0 1 x [ f(t) f(x0 ) ] dt x x 0 x 0 sup f(t) f(x0 ) t between x 0 nd x Since f is continuous t x 0, the right hnd side converges to zero s x x 0. c Joel Feldmn All rights reserved. Jnury 20, 2017 Properties of the Riemnn Stieltjes Integrl 7
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