Page 215 - DMTH401_REAL ANALYSIS

Unit 16: Uniform Convergence and Continuity

16.6 Summary Notes

 If a sequence of functions f (x) defined on D converges uniformly to a function f(x), and if
n
each f (x) is continuous on D, then the limit function f(x) is also continuous on D.
n
 All ingredients will be needed, that f converges uniformly and that each f is continuous.
n n
We want to prove that f is continuous on D.
 Before we continue, we will introduce a new concept that will somewhat simplify our
discussion of uniform convergence, at least in terms of notation: we will use the supremum
of a function to define a ‘norm’ of f.
 A sequence f defined on a set D converges (point wise or uniformly) almost everywhere
n
if there is a set S with Lebesgue measure zero such that f converges (pointwise or uniformly)
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on D\S. We say that f converges (pointwise or uniformly) to f a.e.
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 Much more can be said about convergence and integration if we consider the Lebesgue
integral instead of the Riemann integral. To focus on Lebesgue integration, for example,
we would first define the concept of “convergence almost everywhere”.
 In other words, convergence a.e. means that a sequence converges everywhere except on
a set with measure zero. Since the Lebesgue integral ignores sets of measure zero,
convergence a.e. is ready-made for that type of integration.

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

16.7 Keywords

Convergence almost Everywhere: A sequence f defined on a set D converges (pointwise or
n
uniformly) almost everywhere if there is a set S with Lebesgue measure zero such that f converges
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(point wise or uniformly) on D\S. We say that f converges (pointwise or uniformly) to f a.e.
n
Uniform Convergence Preserves Continuity: If a sequence of functions f (x) defined on D converges
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uniformly to a function f(x), and if each f (x) is continuous on D, then the limit function f(x) is also
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continuous on D.
Lebesgue’s Bounded Convergence Theorem: Let {f } be a sequence of (Lebesgue) integrable functions
n
that converges almost everywhere to a measurable function f. If |f (x)|  g(x) almost everywhere
n
and g is (Lebesgue) integrable, then f is also (Lebesgue) integrable and:
lim f ò n - f dm = 0
n®¥
16.8 Review Questions

n
n
=
x
1. Let f ( ) x on (0, 1). We can observe that f - f = sup x - 0 = 1 ® 0. By
n n sup x (0,1)
0
Theorem f ® uniformly.
n
1, x  n
2. Let f (x) = {0, x < n on . Then f sup = sup x f n ( ) 0 = ® 0. So f ® uniformly.
0
x
-
1
n
n
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