HighDimensional Measures and Geometry
Lecture Notes from Feb 18, 2010
taken by ALI S. KAVRUK
6.2.1 Theorem.
Let
˜
S
n
−
1
=
{
x
∈
R
n
:
°
x
°
=
√
n
}
and
˜
μ
n
be the rotation invariant Borel
probability measure on
˜
S
n
−
1
.Con
s
ide
r
Φ:
˜
S
n
−
1
→
R
given by
Φ(
x
1
,...,x
n
)=
x
1
.
Let
ν
n
be the the probability measure on
R
induced by
Φ
.I
f
γ
1
is the standard Gaussian measure
on
R
then for any Borel set
A
lim
n
→∞
ν
n
(
A
γ
1
(
A
°
A
1
√
2
π
e
−
x
2
/
2
dx
and the density of
ν
n
converges uniformly on compact subsets to
γ
1
.
Proof.
It is enough to show the frst part oF the claim For an open interval
A
=(
a,b
)
. Assume
n
is large. Note that
Φ
−
1
(
A
{
x
∈
R
n
:
a<x
1
<b
}
. So For a fxed
1
,
Φ
−
1
(
{
x
1
}
)
is the
n
−
2
sphere oF radius
±
n
−
x
2
1
which we denote by
˜
S
n
−
2
(
±
n
−
x
2
1
)
.Hence
ν
n
(
A
)=˜
μ
n
(Φ
−
1
(
A
)) =
1
n
(
n
−
1)
/
2

S
n
−
1

²
³´
µ
normalization
°
b
a
¶
¶
¶
¶
˜
S
n
−
2
·
¸
n
−
x
2
1
¹¶
¶
¶
¶
²
³´
µ
(
√
n
−
x
2
1
)
n
−
2

˜
S
n
−
2

º
1+(
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 Spring '08
 Staff
 Geometry, Probability, RK, Compact space, Dominated convergence theorem, rotation invariant Borel

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