Lecture
XXIV
Measures
Remember that an
elementary region
R
in
E
2
is
a region
that has
a simple,
closed,
piecewise
smooth
curve
C
as
its
boundary.
A regular
region
R
is
a region
that is
either regular or can
be divided
into finitely many regular
regions.
In
the
second
case,
the
boundary of
R
consists
of
two or
more simple,
closed,
piecewise
smooth
curves,
of
which
one
is
the outer
boundary curve,
denoted
by
C
,
and
the
others
are
interior boundary
curves
C
1
, . . . , C
n
.
Then
R
consists
of
the points
in
the
interior of
C
,
excluding those in
the interior
of
C
1
, . . . , C
n
.
Similarly,
in
E
3
a region
is
regular
if
its
boundary is
a simple,
closed,
piece
wise
smooth
surface
S
.
An
elementary region
in
E
3
is
either
regular
or
can
be
divided
into finitely many regular
regions. The boundary of
an
elementary
region
that is
not regular is
formed
by one or
more closed,
but not simple piece
wise
smooth
surfaces.
In
the case that the boundary is
formed
by two or
more
surfaces,
one
of
them
is
the
outer
boundary surface
S
,
and
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 Two '04
 Duorg
 Math, Continuous function, Subregion, CN, measure, finite measure

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