The Boole Transformation
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
20 November, 2006
(at
18:39
)
Abstract:
The Boole map preserve Lebesgue measure
on the onepoint compactification
˙
R
of
R
. The scaled Boole
map preserves the finite measure having density 1
/
[
x
2
+ 1].
A condition for an endomorphism of
R
to be
measurepreserving.
Fix an open subset
X
⊂
R
.
When does a continuously differentiable (
not necessar
ily invertible
) map
f
:
X
preserve Lebesgue measure?
(
N.B. For simplicity, we assume henceforth that
f
is every
where finitetoone.
) Evidently
f
is measurepreserving
exactly when
∀
x
∈
X
:
X
r
∈
f
1
(
x
)
1

f
0
(
r
)

=
1
.
For consider a small interval
I
centered at
x
.
The
inverse
images
of
x
,
write
them
r
1
<
. . .
<
r
N
,
have
some
minimum
distance
between
them
and so, if
I
is small enough, then
f
1
(
I
) consists
of
N
disjoint intervals whose lengths are essentially
Len(
I
)
/f
0
(
r
n
)
N
n
=1
.
The intervals have “exactly”
these lengths when
I
is infinitesimal.
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 Fall '09
 Staff
 Probability theory, measure, Lebesgue measure, Lebesgue, Boole Transformation

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