D union disjoint length - UC Santa Barbara
Transcript of D union disjoint length - UC Santa Barbara
Homework 5 Solutions
Katy Craig 2020
Weproceed by induction
Base case n 0 Co Eo D is the union of20 1 disjoint closed interval of length64
Inductive step Assume the result holds
for Cn i By definition Cn is formedfrom Cn i by removing the open middlethird of each interval Thus thenumber of intervals doubles and the
length of each interval is a third ofwhat it used to be Also all of theintervals remain closed and disjoint ThusCn is the union of 2 2n 1 2 n
disjoint closed intervals each of lengthE Esti CIM
First note that part ensuresCnt BIR Hence C E BR E ell zoo
IR 1 X
Also bydefinition CZ Cz Z and
2K Nco D L Thus by continuityfrom above
NC HAICnl nFo Hcn
line.AE HsT tinI.EsY O
First we show f is nondecreasing The problemstats that f is nonde
creasingon G DIC
Thus for all to t c Co DX with tottiIto Ef Iti
d
Fix x ye Co I withxcy.caseix yE0 Dlc ByLtd with tox try
fhdeflyCasey XECO DX and yEC Since XELo BK Kyf Ix E
ypgfit te LO DIC fly
Case3 X EC and yet DK If x O thensince ft 0 for all te fo DIC fade ftpSuppose x 0 By G with tryfor all to c o DIC to y
f Ito E fgSince to ex implies tody
f txt f It tea Dlc Efly
Case 4 exec and yEC
CLAIM There exists toEG Dl C sit XL todyPf of CLAIM Assume for the sake of contradictionthat for all to CCx
yto c C Then C
gontainsan open interval of le.ngth y x o
thus XD Vx yD y x 0 whichcontradicts
By the CLAIM and Cases 2 and 3f x Effto Ef
gNext we show f is continuous
Since f is nondecreasing Jiffy and gTx Kyexist and f is discontinuous at a
point Xo iff ligFxofly C yifxofly If xo O
we use the convention xoflg7 fCo and similarlyif Xo L we take ligIxofly f 1 I
Suppose f is discontinuous at Xo ECO D Then
finkoflyl glifxofly 4 f CEO D
However every open intervalin LEED
contains a point of the form In for some
KEIN n ft 2n D Thus I KEIN n ft 2n Dand X E fo DX sit
fix2k
E finkoflyl glifxofly
This is a contradiction Thus f is
continuous at Xo for all Xo E G D
Since Hx is non decreasing gCx fHtxis strictly increasing hence injective
Since ffx is continuous gtx fCxHx is
continuous By the intermediate value theorem
y
it attains all values in the interval
glo gaD 0,23 so it is surjective
To show g is continuous fix UE Lo B openIt suffices to show g D YU gCU is
open Since K Eo Dl U is compactandg
is continuousgk is compact
Thus the following set is open
0,231 gCK GEO DIK gluThis gives the result
ijective
First note that gktgnhcnlnfgknscn.gsa countable union ofKy
ftp.iiitnii gaingbijective
Since g is continuous gckin is compactThus ga E BR
Now by countable additivity2 260,231 160,23Igcc Nyce
i g i gA Thus it suffices to show 160,2 lg4D 1
To do this note that 8,231gcctgco.DKand recall that co Dic Ii whereIi are disjoint open intervals andf is constant on each interval i e fCIit Ci
Now
Neidigd Algae Dich Ngl tillH go.IE EFffitgiIIiD EHffIDtIil
translation invariance
EE Nci t Ii NII HE Iit No BKcountable additivity
By part b and countable additivity1 NEO D Ko Dlc tx c HEO Blato
Thus No 2 Igcc 160,1314 1 B CHthis gives
the result Y
By HW4 Q5 there exists kid s t
X Amc D 216 d Note ctd
It suffices to show E An Cgdsatisfies that E E contains an open interval
First note that Cc d Cc d containsthe interval l l l where l d c Wewill show f E E EE E This is equivalentto showing that t X Etf E there existsee ez EE s t ee ez X S e Eez TXIn other words it's equivalent to show thatE h E tx 0 for all xefE E
Assume for the sake of contradiction thatthere exists x El E E sit Eh Etx pSince EEE d Et x E c E ITE
Then bymonotonicityand countable additivityZe Nk E dt ET Z NEUE txNE t XEtx 2 XIE Il
4i
This is a contradiction
Recall from HW4 Q6 land as shown inclass there exists AEEO B with the
property that for all ER there is
exactly one YEA sit x ye Q
Also as shown in class A Heth To see this
suppose A Cethos and recall that we showed1 Hits
3 iguana 8 fee HE I
G implies 7CA 0 which contradicts GH
Forany a
beA a b is either 0 or irrationalThus A A does not contain an intervalso the result of part doesn't hold
It suffices to show h Ha to is open
for all a ER By definitionhe Ca t D gf ka tog which is a