3.4 Properties Of The Integral
The integral has many important algebraic properties. First here is a simple
Lemma 3.4.1 Let S be a nonempty set which is bounded above and below. Then if
Proof: Consider 3.4.5. Let x ∈ S. Then −x ≤ sup
follows that −
is a lower bound for
and therefore, −
Now let −x ∈−S.
Then x ∈ S
and so x ≥
is an upper bound for
This shows 3.4.5
. Formula 3.4.6
is similar and is left as an
In particular, the above lemma implies that for Mi
Lemma 3.4.2 If f ∈ R
then −f ∈ R
Proof: The first part of the conclusion of this lemma follows from Theorem
3.3.2 since the function ϕ
is Lipschitz continuous. Now choose P
Then since mi
Thus, since ε is arbitrary,
whenever f ∈ R
and this proves the lemma.
Theorem 3.4.3 The integral is linear,
whenever f,g ∈ R
and α,β ∈ ℝ.
Proof: First note that by Theorem 3.3.1, αf + βg ∈ R
To begin with,
consider the claim that if f,g ∈ R
Let P1,Q1 be such that
Then letting P ≡ P1 ∪ Q1, Lemma 3.1.2 implies
Next note that
For this partition,
This proves 3.4.7 since ε is arbitrary.
It remains to show that
Suppose first that α ≥ 0. Then
If α < 0, then this and Lemma 3.4.2 imply
This proves the theorem.
In the next theorem, suppose F is defined on
Theorem 3.4.4 If f ∈ R
and f ∈ R
, then f ∈ R
Proof: Let P1 be a partition of
be a partition of
Let P ≡ P1 ∪ P2. Then P is a partition of
Thus, f ∈ R
by the Riemann criterion and also for this partition,
Hence by 3.4.9,
which shows that since ε is arbitrary, 3.4.8 holds. This proves the theorem.
Corollary 3.4.5 Let F be continuous and let
be a closed and bounded interval
and suppose that
and that f is a bounded function defined on
which has the property that f is either
or decreasing on
1. Then f
Proof: This follows from Theorem 3.4.4 and Theorem 3.3.2.
The symbol, ∫
when a > b
has not yet been defined.
Definition 3.4.6 Let
be an interval and let f ∈ R
Note that with this definition,
Theorem 3.4.7 Assuming all the integrals make sense,
Proof: This follows from Theorem 3.4.4 and Definition 3.4.6. For example,
Then from Theorem 3.4.4,
and so by Definition 3.4.6,
dF || = ∫
| || = ∫
dF + ∫
The other cases are similar.
The following properties of the integral have either been established or they follow
quickly from what has been shown so far.
The only one of these claims which may not be completely obvious is the last one. To
show this one, note that
Therefore, by 3.4.14 and 3.4.12, if a < b,
If b < a then the above inequality holds with a and b switched. This implies