Page 213 - DMTH401_REAL ANALYSIS

Unit 16: Uniform Convergence and Continuity

Notes
  
x
x
f - f sup < Þ f n ( )  f ( )  f n ( )+ , x
"
x
n
4(b a ) 4(b a ) 4(b a )
-
-
-
i.e.

x
f
x
sup ( )  sup f n ( )+ .
-
[ i t - 1 , i t ] [ i t - 1 , i t ] 4(b a )
Then
n n æ  ö
P
x
f
x
f
U ( , ) = å sup ( )(t - t i- 1 )  å ç sup f n ( )+ ÷ (t - t i- 1 ) =
i
i
-
i= 1 [ i t - 1 , i t ] i= 1 è [ i t - 1 , i t ] 4(b a )ø
 n 
P
f
f
= U ( , )+ å (t - t 1 ) U ( , )+ .
=
P
n
n
i
4(b a ) i= 1 i- 4
-
Similarly,

f
L ( , )  L ( , )- .
P
f
P
n
4
So
    
-
P
f
f
f
f
U ( , ) L ( , ) U ( , )+ - L ( , )+  + + = 
P
P

P
n n
4 4 2 4 4
b b
x
dx
x
Claim 2: a ò f n ( )dx ® a ò f ( ) .
).
-
Since f - f sup ® 0, given  > there exists N   s.t. n"  N : f - f sup <  /(b a Therefore,
0
n
n
" n  N :
b b b b
x
x
x
a ò f n ( )dx - a ò f ( )dx = a ò f n ( )dx  a ò f n ( )- f ( ) dx 
x
x
b 
-
a ò  dx = f - f sup (b a < (b a ) (b a =  .
)
)
-
f
f -
n
n
sup
-
number
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”:
16.4 Convergence almost Everywhere
A sequence f defined on a set D converges (pointwise or uniformly) almost everywhere if there
n
is a set S with Lebesgue measure zero such that f converges (pointwise or uniformly) on D\S.
n
We say that f converges (point wise or uniformly) to f a.e.
n
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.
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