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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
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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
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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
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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
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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