Define the
Lower Sum
L
P
(
f
)
and
Upper Sum
U
P
(
f
)
, as
L
P
(
f
) =
N
i
=
1
m
i
(
P
) (
x
i

x
i

1
)
U
P
(
f
) =
N
i
=
1
M
i
(
P
) (
x
i

x
i

1
)
The idea is that the “area under the graph” is underestimated by
L
P
(
f
)
and
overestimated by
U
P
(
f
)
.
Upper and Lower Sums
U
P
(
f
) =
∑
N
i
=
1
M
i
(
P
) (
x
i

x
i

1
)
L
P
(
f
) =
∑
N
i
=
1
m
i
(
P
) (
x
i

x
i

1
)
Since
m
i
(
P
)
≤
M
i
(
P
)
, clearly
L
P
(
f
)
≤
U
P
(
f
)
for any partition
P
.
Refining the sums
Lemma
If
P
⊂
Q
(
Q
is a refinement of
P
,) then
L
P
(
f
)
≤
L
Q
(
f
)
≤
U
Q
(
f
)
≤
U
P
(
f
)
.
Refining the sums
Lemma
If
P
⊂
Q
(
Q
is a refinement of
P
,) then
L
P
(
f
)
≤
L
Q
(
f
)
≤
U
Q
(
f
)
≤
U
P
(
f
)
.
Let’s assume that
Q
adds just one point
y
to
P
, and
x
i

1
<
y
<
x
i
.
Since
[
x
i

1
y
]
⊂
[
x
i

1
x
i
]
and
[
y x
i
]
⊂
[
x
i

1
x
i
]
we have
M
i

:=
sup
x
∈
[
x
i

1
y
]
f
(
x
)
≤
sup
x
∈
[
x
i

1
x
i
]
f
(
x
) =
M
i
(
P
)
and
M
i
+
:=
sup
x
∈
[
y x
i
]
f
(
x
)
≤
sup
x
∈
[
x
i

1
x
i
]
f
(
x
) =
M
i
(
P
)
The only term in
U
P
(
f
)
which is modified in making
U
Q
(
f
)
is:
M
i
(
P
)(
x
i

x
i

1
) =
M
i
(
P
)(
y

x
i

1
) +
M
i
(
P
)(
x
i

y
)
≥
M
i

(
y

x
i

1
) +
M
i
+
(
x
i

y
)
and so
U
P
(
f
)
≥
U
Q
(
f
)
. If
Q
contains more than one point more than
P
, do this
repeatedly. The opposite happens when we look at the inf,
L
P
(
f
)
Further refinements
Any lower sum is smaller than any upper sum, regardless of partition:
Lemma
Let
f
be bounded and
P Q
be any two partitions of
[
a b
]
. Then
L
P
(
f
)
≤
U
Q
(
f
)
.
Proof: Recall that
P
∪
Q
is a common refinement for both partitions,
P
⊂
P
∪
Q
and
Q
⊂
P
∪
Q
. So we apply the previous Lemma, and
L
P
(
f
)
≤
L
P
∪
Q
(
f
)
≤
U
P
∪
Q
(
f
)
≤
U
Q
(
f
)
♦
Let’s optimize this.
I
For any fixed
Q
,
U
Q
(
f
)
is an upper bound for the
L
P
(
f
)
, over all partitions
P
.
I
For any fixed
P
,
U
P
(
f
)
is a lower bound for the
L
Q
(
f
)
, over all partitions
Q
.
I
Therefore:
sup
{
L
P
(
f
)

P
is a partition of
[
a b
]
} ≤
inf
{
U
Q
(
f
)

Q
is a partition of
[
a b
]
}
Riemann Integrability
Definition
Let
f
be a bounded function.
f
is
Riemann integrable
on
[
a b
]
if
sup
{
L
P
(
f
)

P
is a partition of
[
a b
]
}
= inf
{
U
Q
(
f
)

Q
is a partition of
[
a b
]
}
.
For a Riemann integrable function, we define the Riemann (definite) integral,
ˆ
b
a
f
(
x
)
dx
= sup
P
L
P
(
f
) = inf
Q
U
Q
(
f
)
Is any function
not
Riemann integrable?
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 Fall '13
 Integrals, Riemann integral, Riemann sum, Riemann