Page 214 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
Example: Let r be the (countable) set of rational numbers inside the interval [0, 1],
n
ordered in some way, and define the functions
ì 1 if x = r ,r ,r ,...r n 1 ì if x = rational
1
3
2
í and g(x) í
î 0 otherwise î 0 if x = irrational
Show the following:
 The sequence g converges pointwise to g but the sequence of Riemann integrals of g does
n n
not converge to the Riemann integral of g.
 The sequence g converges a.e. to zero and so does the sequence of Lebesgue integrals of g.
n n
Solution:
Each g is continuous except for finitely many points of discontinuity. But then each g is integrable
n n
and it is easy to see that
ò gn(x) dx = 0
But the limit function is not Riemann-integrable and hence the sequence of Riemann integrals
does not converge to the Riemann integral of the limit function.
Please note that while each g is continuous except for finitely many points, the limit function g
n
is discontinuous everywhere
On the other hand, each g is zero except on a set of measure zero, and so is the limit function.
n
Thus, using Lebesgue integration we have that all integrals evaluate to zero. But then, in particular,
the sequence of Lebesgue integrals of g converge to the Lebesgue integral of g.
n
There are many theorems relating convergence almost everywhere to the theory of Lebesgue
integration. They are too involved to prove at our level but they would certainly be on the
agenda in a graduate course on Real Analysis. For us we will be content stating, without proof,
one of the major theorems.

16.5 Lebesgue’s Bounded Convergence Theorem

Let {f } be a sequence of (Lebesgue) integrable functions that converges almost everywhere to a
n
measurable function f. If |f (x)|  g(x) almost everywhere and g is (Lebesgue) integrable, then
n
f is also (Lebesgue) integrable and:
lim f ò n - f dm = 0
n®¥
Self Assessment

Fill in the blanks:
1. Let I =  and ( ) sin( ).f x = x Then .........................
x
2. Let I = [0,1] and ( )f x = - 2 . Then ...................... The norm stays the same even if we change
the interval [0, 1] to (0, 1).
3. Let { }f ¥ be a sequence of real-valued functions on I. Then f ® f uniformly if and only
n n= 1 n
if ............................ Note that f - f sup is just a sequence of number.
n
4. Let { }f ¥ be a sequence of real-valued functions on [ , ].a b If all f are Riemann-integrable
n n= 1 n
and f ® f uniformly then f is ....................... and
n
b b
dx
x
x
f ( )dx ® f ( ) .
ò a n ò a
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