Remember LP :

LP or,Tsn) = { feels

,8.y) : fl fl Pdn cos} He - null sets

We've looked carefully at Lt and L2.

More generally :

Theorem : 117.241 For l speos ,

Hf Rp : = ( ful fl Pda )"P

defines a norm onLP Irish) .

In particular :

H ft g Hip S H f the t 11g the

t.ge LP .

Thus LP is a normed vector space .

This is Minkowski's inequality .

We proved it already for p- 12 .

In general , it goes beyond the scope of what we need to prove it .

LP us .

La convergenceLemma : Let fn

, ft LMR,qµ) l l speos ) with Hfn - flip → o .

Then fn→µ f .

Pf . Morbius sina.gl?jt:eI..g*nppqi=Iiifn.fnu → o.

¥: is:": 't.ci.

.

tf lfn He} s an!ffn Ian} =L→ o

.

i. fn→go .

pH

Hfn - other = ftp.n/Pdx--&uPdA--ht-hP--nM.ipfn-olkp-nHpto

Theorem : 117.271 For Isp cos , Lt is Cauchy complete :

fine LP,Hfn - fmllhp To as him → as

⇒ I felt sit . llfn - f the → o .

Pf . H - Cauchy ⇒ Le - Cauchy : →

ME lfn - false}"Em tgpflfn-fmp.de = tlfmfmlkp → e

i.

is E - Cauchy .

E' '

names .

€2 fu .→ f as . , fate .Hfn . - flip Ets - ff lfmifnjlpefr Fasten him!flfna-fnjldnkm.lt) s fnssbggiiptllfn

.

- find. .

LIFT " Kiki the

i. It fn-ftkp-Nfn-fnnltlfna-fdlkpsnfn-fnalkptllfna-flkpg-g.skSo as n, knees

Summary[ convergence

"" iii.✓ convergence

DCTJ

Ip > t) 2- as. convergence

Theorem : ( Lo - L'DCT)

Let fn.gn.ge Lt , fell , s- k

( D lfnlsgn a. s .

(2) f.→af , gn→µg

③ Ign → fgThen ft L' and stfu → e .

( In particular, ffn -off . )Pf. By al , can find fun→ f Agha→g a - s

.

If Isa .by?.lfnalsfF.gne--g a. s .

S -

pose thatf#→ .

Fao, 8 me sit

.ftVk ,

But fun -7 f,i . fna→f as . gna → g a - s .

If - finals Ifl tlfnnlsgtgn.→ 2g . Also flgtgmd '- fgtfgmeosfzg .

i. by DCT ⇒ lff . Nu .i. I Sfnisfl -- Isan -Al sflfnfto .H