The Riemann Integral

16 1. The Riemann Integral
Similarly, L(cf;P) = cL(f;P) and L(cf) = cL(f). If f is integrable, then
which shows that cf is integrable and
we have
Now consider −f. Since
Therefore
U(cf) = cU(f) = cL(f) = L(cf),
∫ b
a
sup(−f) = −inf f,
A A
U(−f;P) = −L(f;P),
cf = c
∫ b
a
inf
A
f.
(−f) = −supf,
A
L(−f;P) = −U(f;P).
U(−f) = inf U(−f;P) = inf [−L(f;P)] = − sup L(f;P) = −L(f),
P∈Π P∈Π P∈Π
U(f;P) = −U(f).
L(−f) = sup L(−f;P) = sup
P∈Π P∈Π
Hence, −f is integrable if f is integrable and
∫ b
a
[−U(f;P)] = − inf
P∈Π
(−f) = −
Finally, if c < 0, then c = −|c|, and a successive application of the previous results
shows that cf is integrable with ∫ b
a cf = c∫ b
a f.
□
Next, we prove the linearity of the integral with respect to sums. If f, g are
bounded, then f +g is bounded and
It follows that
sup
I
(f +g) ≤ sup
I
f +sup
I
∫ b
a
f.
g, inf(f +g) ≥ inf f +infg.
I I I
osc(f +g) ≤ osc f +oscg,
I I I
so f+g is integrable if f, g are integrable. In general, however, the upper (or lower)
sum of f +g needn’t be the sum of the corresponding upper (or lower) sums of f
and g. As a result, we don’t get
∫ b
a
(f +g) =
simply by adding upper and lower sums. Instead, we prove this equality by estimating
the upper and lower integrals of f +g from above and below by those of f
and g.
Theorem 1.24. If f,g : [a,b] → R are integrablefunctions, then f+g is integrable,
and
∫ b
a
(f +g) =
∫ b
a
∫ b
a
f +
f +
∫ b
a
∫ b
a
g
g.