350
[VoL
6,
78.
A
Converse
of
Lebesgue’s
Density
Theorem.
By
Shunji
KAMETANI.
Taga
Higher
Technical
School,
Ibaragi.
(Comm.
by
S.
KA,
M.I.A.,
Oct
12,
1940.)
I.
The
main
object
of
the
present
note
is
to
see
that
a
converse
of
Lebesgue’s
density
theorem
holds.
We
shall
consider,
for
brevity,
sets
of
points
in
a
Euclidean
plane
R
s
only.
But
the
results
which
will
be
obtained
can
obviously
be
ex
tended
to
spaces
R
of
any
number
of
dimensions.
The
Lebesgue
outer
measure
of
a
set
E
in
R
will
be
denoted
by
EI.
Let
z
be a
point
of
R
and
Q
an
arbitrary
closed
square
contain
ing
z
with
sides
parallel
to
the
coordinateaxes.
We
shall
denote
by
](z,
E)
and
D(z,
E)
the
superior
and
the
in
ferior
limit
respectively
of
the
ratio
QEI/I
Q]
as
the
diameter
of
Q
tends
to
0
or
]Qi0,
and
shall
call
them
the
upper
and
the
lower
density
of
E
at
z
respectively.
If
they
are
equal
to
each
other
at
then
the
common
value
will
be
called
the
density
of
E
at
z.
The
points
at
which
the
density
of
E
are
equal
to
0
are
termed
points
of
dispersion
for
E.
It
is
well
known
by
Lebesgue’s
density
theorem
that,
f
set
of
points
are
measurable,
almost
every
point
of
its
complementary
set
is
a
poin
of
dispersion
for
he
given
set.
D
II.
We
shall
prove
the
following
theorem
which
evidently
contains
a
converse
of
the
above
proposition.
Theorem
1.
Let
E
be
a
pointset
whose
lower
density
is
0
at
almos
every
point
of
the
complementary
se
of
E.
Then
the
se
E
is
mrable.
Proof.
We
can
obviously
assume,
without
loss
of
generality,
that
the
set
E
is
bounded.
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 Spring '10
 Student
 Euclidean space, measure, Lebesgue measure, Lebesgue integration

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