MATH 321:201: Real Variables II (Term 2, 2010)
Home work assignment
Due date: Friday, Jan 15, 2010 (hand-in in class)
Problem 1 : Find a monotonically increasing function α on the unit interval [0, 1] such that
R1
R1
(0)
• for any continuous function f : [0, 1] → R, 0 f dα = f (1)+f
+ 0 f dx;
2
• α(0) = 0.
Prove why your choice of α works.
Problem 2: Do [Rudin, Ch. 6. Exercise 2].
Problem 3 : (Summation by parts)
Let {fi }, {αi } be two sequences of real numbers. Denote ∆fi = fi − fi−1 , ∆αi = αi − αi−1 .
Show
n
n
X
X
fi ∆αi = −
αi−1 ∆fi + fn αn − f0 α0 .
i=1
i
Problem 4 : (Integration by parts)
Let f, α be two bounded functions on the interval [a, b], which are both monotonically increasing:
i.e. f (x) ≥ f (y) , α(x) ≥ α(y) for x ≥ y in [a, b]. Show the following:
•
f ∈ R(α) ⇐⇒ α ∈ R(f ),
i.e. f is (Riemann-Stieltjes) integrable with respect to α if and only if α is (RiemannStieltjes) integrable with respect to f ;
• Moreover, if f ∈ R(α) (so α ∈ R(f )), then
Z b
Z b
αdf + f (b)α(b) − f (a)α(a) .
f dα = −
a
a
Recommended exercises (Please DO NOT hand-in) : Rudin, Ch. 6, Exercises # 1, # 3, # 5, # 6