Page 282 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Therefore, with the help of (28), applied to partitions of [a, x], (f )  (g ) = g and h =
+
+
i i i i
–
–
(–h )  (f ) .
i i
Hence P (f, [a, x])  P (g, [a, x]) = g(x) – g(a). Similarly, h(x) – h(a) = N (–h, [a, x])  N (f, [a, x]).
   
When, in each case, we take the supremum over all |[a, x], we get P(x)  g(x) – g(a) and N(x)  h(x)
– h(a). These ”say” that there is no “wasted cancellation” in the formula f(x) = f(a) + P(x) – N(x).
Task Prove that if f(x) Î BV[a, b] and f(x) is continuous at x Î [a, b] then so are P(x), N(x)
o
and V(x).
Self Assessment

Fill in the blanks:
1. A ..................................... f(x) defined on the bounded and closed interval [a, b] is Riemann-
Stieltjes integrable on [a, b] with respect to  if there exists a number RSI such that for all
 > 0 there exists  > 0 such that for every partition  of [a, b],
mesh() <   |RS(f, ) – RSI| < .
2. If f is Riemann integrable on [a, b] then f is .................................
3. If (x) is defined for x Î [a, b], we denote by  = (, [a, b]) the set of all c Î [a, b] such that
every open interval U that contains c contains x < c < x with .....................................
1 2
4. If f is ........................................... on [a, b] with respect to  then f is bounded on (, [a, b]).
5. If f and  have a ................................... at the same point, then the Riemann-Stieltjes integral
does not exist.

22.7 Summary

 A Riemann-Stieltjes sum for a function f(x) defined on an interval [a, b] is formed with the
help of
(a) A partition  of [a, b], namely an ordered, finite set of points x, with a = x < x < <
i 0 1
x = b (where n is a positive integer that can be any positive integer, and one that we
n
will often write as n = n ),

(b) A selection vector  = ( , ...,  ) that has n components that must satisfy x    x ,
1 n  i – 1 i i
for i = 1, 2, ..., n. and
(c) An integrator (x), which is a function defined on [a, b] that plays the role of the x in
dx ...
A Riemann-Stieltjes sum for f over [a, b] with respect to the partition , using the
selection vector , and integrator , may be denoted (in greatest detail!) as follows,
and it is given by the value of the sum following it:
n 
(d) RS (f, , [a, b], , ):= å f( )( (x ) -  (x i 1 )) .


i
i
-
i 1
=
We try to allow context to let us drop some of the items inside the RS(...).
 In this definition, as in the Riemann-sums definition, we can write x := x —x or  :=
i i i – 1 i
(x ) –(x ). These are convenient because they are short and suggest the dx or d in an
i i – 1
integral. But they can cause confusion because they leave out the dependence they have on
x . The x is used in the Riemann-Stieltjes context.
i – 1 i
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