Page 250 - DMTH401_REAL ANALYSIS

Real Analysis

Notes We claim that the sets of upper and lower sums corresponding to different partitions of [a,b] are
bounded. Indeed, let m and M be the infimum and supremum of f in [a.b].

Then m £ m £ M £ M and so
i i
m  x £ m  x £ M  x £ M  x i
i
i
i
i
1
Putting i = 1, 2, ........ n and adding, we get
n n
£
m å  x, L(P,f) U(P,f) M å A x,.
£
£
i 1 i 1
=
=
n n
å  x = å (x - x i 1 ) = x - x = b a
-
-
i
n
0
i
=
=
i 1 i 1
Thus m(b a) L(P,f) U(P,f) M(b a)- £ £ £ -
For every partition P, there is a lower sum and there is an upper sum. The above inequalities
show that the set of lower sums and the set of upper sums are bounded, so that their supremum
and infimum exist. In particular, the set of upper sums have an infimum and the set of lower
sums have a supremum. This leads us to concepts of upper and lower in tegrals as given by
Riemann and popularly known as Upper and Lower Riemann Integrals.
Definition 5: Upper and Lower Riemann Integral
Let f: [a,b]  R be a bounded function. The infimum or the greatest lower bound of, the set of all
upper sums is called the upper (Riemann) integral o f f on [a,b] and is denoted by,
h
 f(x)dx.
a
i.e.
b
=
 f(x)dx. g.l.b. {U(P,f): P is a partition of [a,b]}.
a
The supremum or the least upper bound of the set of all lower sums is called the lower (Riemann)
integral of f on [a,b] and is denoted by
b
 f(x)dx
a
i.e.

b
 f(x)dx = l.u.b {L(P,f): P is a partition of [a,b]}.
a
Now we consider some examples where we calculate upper and lower integrals.

Example: Calculate the upper and lower integrals of the function f defined in [a, b] as
follows:
1 when x is rational
f(x) =
0 when x is irrational
Solution: Let P = {x , x … x } be any partition of [a,b]. Let M and m be respectively the sup. f and
0 1 n i 1
inf. f in [x , x,]. You know that every interval contains infinitely many rational as well as
i-1
irrational numbers. Therefore, m = 0 and M, = I for i = 1, 2 ... n. Let us find U(P,f) and L(P,f).
i

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