Théorème de Prokhorov, théorème de Lévy

ð
pE, dqM1pEqE
ΓĂM1pEqą0
KĂE µ PΓµpKq ě 1´
pE, dq pµnq
M1pEq pµnq
ð pµnq
pě1KpĂE n ě0
µnpKpq ě 1´1
p.
KpCpKpqpě1
pgp,kqkě0fk,p PCpEq
gk,p
pk, pq pµnpfk,pqqnr˘}fk,p}8s
φ k, p
pµφpnqpfk,pqqn
fPCpEq pµφpnqpfqqn
ą0pě11
p}f}8ď k }gk,p ´f1Kp} ď 

pgp,kqkě0CpKpq }fk,p}“}gk,p} ď }f} ` 
pµφpnqpfk,pqqnN n, ˜něN
ˇˇµφpnqpfk,pq ´ µφp˜nqpfk,pqˇˇď.
n, ˜něN
ˇˇµφpnqpfq ´ µφp˜nqpfqˇˇď |µφpnqpf1Kc
pq| ` |µφpnqppf´fk,pq1Kpq| ` |µφpnqpfk,p1Kc pq| ` p ˜nq
` |µφpnqpfk,pq ´ µφp˜nqpfk,pq|
ď2p`` p}f} ` q1
pq ` 
pµφpnqpfqqn`pfq
`|`pfq| “ lim |µφpnqpfq| ď }f}fě0
`pfq “ lim µφpnqpfq ě 0µ
`“µp¨q
ñ pµnq
M1pEq
G E νnν
lim inf
nÑ8 νnpGq ě νpGq.
fk“minp1, kdp¨, Gcqq
1G
k
lim inf
nÑ8 νnpGq ě lim inf nÑ8 νnpfkq “ νpfkq.
νpfkq Ñ
kÑ8 νpGq
pxpqE k ě1
ď
p
Bpxp,1{kq “ E.
Gq,k :“
q
ď
p“1
Bpxp,1{kq
ą0kě1qk, ně1
µnpGqk, ,kq ě 1´.
M1pEq
Ξtµn, n ě0u
fq: Ξ Ñ r0,1s
µÞÑ µpGk,q q.
1
Ξ

0ą0k0ě1
qrνrtµn, n ě0ur
νrpGqr,k0q ď 1´0.
νr
νrν
r
νpGqr,k0q ď lim inf
˜rÑ8 ν˜rpGqr,k0q ď 1´0.
r`8
νpEq ď 1´0,
ą0kě1
qkn
µn˜qk
ď
p“1
Bpxp,1{kq¸ě1´
2k.
K:“č
kě1
qk
ď
p“1
¯
Bpxp,1{kq
1´řkě1{2k“1´ µn
E K
Rd
µ ν Rd
fRd
µpfq “ νpfqą0Aą0
µpr´A, Asdq ě 1´, νpr´A, Asdq ě 1´,
fr´A, Asd
2A P d
supr´A,Asd}f´P}8ď µ ν
µpPq “ νpPq
|µpfq ´ νpfq| ď |µpfq ´ µpPq| ` |νpfq ´ νpPq|
ď2 sup
r´A,Asd
}f´P}8`µp|P|1pr´A,Asdqcq ` νp|P|1pr´A,Asdqcq,
ď` }P}8µppr´A, Asdqcq,
ď` p}f}8`q.
µpfq “ νpfq

XnRd
φnφ0
φ X XnX
pXnq
uą0
1
udżr´u,usd
p1´φnptqq t“E«1
udżr´u,usd
p1´ext,Xnyqtff
“2dE«1´
d
ź
i“1
sinpuXpiq
nq
uXpiq
nff
ě2dE«1´
d
ź
i“1ˇˇˇˇˇ
sinpuXpiq
nq
uXpiq
nˇˇˇˇˇff
ě2dˆ1´1d´11
2˙
looooooomooooooon
“1{2
´Di:|Xpiq
n| ě 2{u¯,
sˇˇ
sin s
sˇˇď1|s| ě 2ˇˇ
sin s
sˇˇď1{2
´Di:|Xpiq
n| ě 2{u¯ď1
udżr´u,usd
p1´φnptqq tď1
udżr´u,usd
|1´φnptq| t.
lim sup
nÑ8 ´Di:|Xpiq
n| ě 2{u¯ď1
udżr´u,usd
|1´φptq| t.
φ0
1
udżr´u,usd
|1´φptq| tÝÑ
uÑ00.
lim sup
uÑ0
lim sup
nÑ8 ´Di:|Xpiq
n| ě 2{u¯“0,
lim sup
MÑ8
lim sup
nÑ8
p}Xn} ą Mq “ 0 :
pXnq