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18.102 Introduction to Functional Analysis
Spring 2009
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18
LECTURE
NOTES
FOR
18.102,
SPRING
2009
Lecture
4.
Thursday,
12
Feb
I
talked
about
step
functions,
then
the
covering
lemmas
which
are
the
basis
of
the
definition
of
Lebesgue
measure
–
which
we
will
do
after
the
integral
–
then
properties
of
monotone
sequences
of
step
functions.
To
be
definite,
but
brief,
by
an
interval
we
will
mean
[
a, b
)
–
an
interval
closed
on
the
left
and
open
on
the
right
–
at
least
for
a
little
while.
This
is
just
so
the
length
of
the
interval,
b
−
a,
only
vanishes
when
the
interval
is
empty
(not
true
for
closed
intervals
of
course)
and
also
so
that
we
can
decompose
an
interval,
in
this
sense,
into
two
disjoint
intervals
by
choosing
any
interior
point:
(4.1)
[
a, b
) = [
a, t
)
∪
[
t, b
)
,
a < t < b.
Now,
by
a
step
function
(4.2)
f
:
R
−→
C
(although
often
we
will
restrict
to
functions
with
real
values)
we
mean
a
function
which
vanishes
outside
a
finite
union
of
disjoint
‘intervals’
and
is
constant
on
each
of
them.
Thus
f
(
R
)
is
finite
–
the
function
only
takes
finitely
many
values
–
and
(4.3)
f
−
1
(
c
)
is
a
finite
union
of
disjoint
intervals
, c
= 0
�
.
It
is
also
often
convenient
to
write
a
step
function
as
a
sum
N
(4.4)
f
=
c
i
χ
[
a
i
,b
i
)
i
=1
of
multiples
of
the
characteristic
functions
of
our
intervals.
Note
that
such
a
‘pre
sentation’
is
not
unique
but
can
be
made
so
by
demanding
that
the
intervals
be
disjoint
and
‘maximal’
–
so
f
is
does
not
take
the
same
value
on
two
intervals
with
a
common
endpoint.
Now,
a
constant
multiple
of
a
step
function
is
a
step
function
and
so
is
the
sum
of
two
step
functions
–
clearly
the
range
is
finite.
Really
this
reduces
to
checking
that
the
difference
[
a, b
)
\
[
a
�
, b
�
)
and
the
union
of
two
intervals
is
always
a
union
of
intervals.
The
absolute
value
of
a
step
function
is
also
a
step
function.
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 Spring '09
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 Derivative, step functions, sj

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