Homework 6 Solutions

KatyCraig 2020

First note thatch E E IE EMR EBR is a o algebraIf A tool then Acn IR CARVE to UE ISNA AIR

IRIA Rl AMIR C Bits since A AIRE BIRand co algebras are closed under complementsIf A i3 Eck then 7ADAIR i CA in IR CBIRSince Ain R E BR ti EN and o algebrasare closed under countable unions

ByHW4 Q2 Bpi is generated by setsof the form la ta to and E 03 a EIRSince all of these sets belong to UBEE cel

Conversely if E Eek then E h IR EB IR andsince Baz is generated by Ca ta aeR3E h IR Ect Ela to aeR3 ThusEE ch CECaito AER Eto Eof BE This showsell C Bpi

O

First suppose f is nondecreasing

It suffices to show that f Etd f YETand f Ca ta are measurable for all aelRsince such sets generate BR

r respf CE a

Note that f to is a Bosisintermital in IRsince if X YE f

toKy then for all

2 Efx gf non decreasing ensures f z tag

resp flz D

Similarly for all AER f ka th is alsoaPissibineteifval in IR since if X y

tf ka th

Ky then f tx a and fly a sofor all 2 cCx

gfCz a

Since all intervals including the empty set

belong to Bite this gives the result forf nondecreasing

Now suppose f is nonincreasing Thenf f 1 C f Since I 17 and f f arenon decreasing they are measurable Finallythe product of measurable fns is measurable

For all r 0 Bix r nIBfx r th SinceBix r InBE is a sequence of nested setsbycontinuityfrom belowµ BK r LinforcBtx r LD G

Consider a sequence Xm Xo Then forall new there exists Mst

Ix m Xol E tn V m Z M

Thus BCxo r HEB Xm r t m z mTherefore

ngonotonicity

limits µ Bhim r 2µL B ko r tn

since NEIN was arbitrary by we obtainmm µCBtxmrHZµlBlxo rt

This shows xt HCBtx.nl is lowersemicts

For all r o BTx.rt I.is x rttnBy continuity from above since µ is locallyfinite B Ix rf info B fx r th

Consider a sequence Xm xo As above tnHF M sit

B him r C BExo r t th t m M

As above

mm BMCBTxm.rs Eu Bha rttn

and since NEIN was arbitrarymm BoulBTxm r FulBTxo rt

Thus xHµCBTx rn is upper semi continuous

By definition of Kmsupf txt fief Gr tx

By H W 5 0 5 Gr tx is lower semi continuousBy HW4 Q2 this implies it is Borelmeasurable

Finally LiaoGrhd info Gynhd where thelimit exists bymonotonicity Thus Axlis a sequence of measurable functionshence measurable