Math 617
Doig
:
Gnvgmethni
Dominated
li i ) F
f
Then
fix
maske
-
e
at he
henry
ELYXM
lit ( XM,
.
fdu
g : Hu
A- i
th
an
fnlx )
Gsidr
the fn
Then
X
=
ffndm
{
flns
n
) st
I
.
fnltg
if
infinity
DCT
Proof of
,
by
We
:
duty
which
you
xecn
my
the obvisf µtgnk
,
ipks
-
g
if
follow
Now
z
,
an
A
.
Apply
±
f.
re
E
we
Y the 2
Qiwht
do
Fatos
lemm
ffdutfgdn
=
.
0
to
wie
point
.
Bit obvioyf
.
ht
.
f
&
sqrekf
.
we
Talons
can't be
6mm says
so
tht
might
as
fist
well
an
qnlif
cannot crate
you
Alf
the fn &
f
,
-
a
pshy
by
in the
mss
lmthrtnu
pmtuie
assure
Essen
.
limit
e.)
.
Ifnkgewyahe
&
real ukd
-
are
it off to
,
so
-
,
htf
line
the
fn+g20
sums
Sf
hnxntj
case
du E
.
&
Sfdm
g.
Sfdu
fit
0
.
±
so
&
light ffndu
the
are
) fndu
knssyp
2
do exist
,
& the
.
gurus
f
nanny
.tk frothy
.
?
!
flft
so
.
g) du
get
,
±g ,
.
hphy tht
g
fn
team
Talons
the limit
pt
Ef kg
show
can
the IN
.
÷o
in
mss
Sfn du exib
it
hw
yet
't
ky
We of
.
&
ftp.sfndn-1
so
,
with red
Ringing
obviously
.e
:|
s
:D
in
will mdity the fn
as
a
f intgnfleffs
mu
.
else
o
&
destroy
in
-
in
be
e
:X
-
.
.
Note tht His exapleshstht the kywlitfn
.
÷
.
,
g?
1
th
I
=
.
.e
a
ni
fi fu
& ht
spa
mere
a
f
-
oh
no
,
nµ
Siple
a
is
=
,n,
be
him S fndn
=
fail if the
,
)
f.
.
GELYXM
:
,
st
m
=
ftp.lfntgdn
Sfdm El ,y÷f
ff
.
I
du
lmnisnfo
Slfntgdu
=
hwnynfoffndu
+
Sg du
result
will
.
gdn
as
tinmnupsfndu
ffdm
.
so
-
=
.
.
Sfdu
.
So combing these g
fly Adu Sky (g- fdduthmmnof
-
E
-
linmssuopffndn
Sfdmlftdnt link
inn
:p
Sfndn
=
fgdu
Effdn
lnmggpffndu
or
;
Hdn
S ( g-
-
.
.
,
the poof
completing
Prep
If
:
f
1
is
R
3
f
Riemann
intake
obvious
givnhw
B
hopefully
Rian
,
Gwgu=
fan
we
1
2
3
4
f
then
:
#
( {xi
fdiknlinwsdx } )
defined
the lehesguintgnl
m
we
lebesguntgnbk
B
bit ntuntmf
points
ytd the
an
:
denial analysis abut point wise
cnwsy
Bt
mtglh
th
.
list with
fn
army
fu
on
fn
fn
.
to
.
!
S fkldxt §yjtdm
&
.
.
is
2
hdr
,
ht
mtituf sort
hopefully
dear
of
.
:
Types of
we hw
bad
>
Tf f
2
O
Cats )
:
Hkgahtn add the bksgumtgd
the irtgnl wire
XHR ,
When
#D"
.
converges
converges
to
0
f
pomhie
to
f
uniting
causerie
of
f pmlwieahrtemynhe
to
atone
,
ad
buff thnenbrexapbs
the
flx)
=
th
,
uhih
.
other types
to
nry
to
may
& uniform
anwgue
f unify ahtergnhe (
if
fly ) for
fly
essnlny
or
:
unify
or
µ
.
into
a
.e
xex
.
nom
.
) if
KE
> 0
NEIN
J
sit
nzn
/ ftp.flxy ,
-
5
6
7
fn
fu
f.
onvyy to
anvyy
f
KE
if
Hfn
f
to
cmvyy to
admit nnifmf if
f
in
in
L
'
music
mm
if
HE
> 0
>
-
,
0
fllu
I
=
EEMY
µ( E)
Slfn fldn
0
'
if { xexilklxl
-
EE
as
HAIZE } )
st
n
-
fn
.
•
0
unvgy
miff
.
as
no
.
to
f
in
It
^
,
{
"
e
.
Note
,
In
prhnbihty they
ad 70
.
}
albd
the In
4
f
4
with )
mdm
,
convergence in
probability
.
6
Balled
convergence
in
mean
,
3
is
albd
almost
sue
convene ,