It is strongly recommended that …
It is strongly recommended that you refresh the basic theory of measure and integration. This may be found in Teschl Ch.7 and 8 (MAT 4400/3400) or in Folland, Real Analysis, Ch. 1 and 2. We list below some useful exercises.
In Teschl Ch. 7:
Problem 2, 5, 10, 11, 13, 14.
In Folland: Exercise 2.28
Further exercises:
Exercise 1.
(a) Let f be a nonnegative function on a measure space (X,M,?). How is the Lebesgue integral of f defined?
(b) Let X= [a,b], M = the Borel σ-algebra on [a,b], ?= the Lebesgue measure on M. Explain that for step-functions s on [a,b] the Lebesgue integral is equal to the Riemann integral. Also explain that the Lebesgue integral of a continuous, non negative function f is equal to its Riemann integral. (The Dominated Convergence Theorem could be useful.)
Exercise 2.
Derive the Monotone Convergence Theorem from Fatou's Lemma.
Exercise 3.
Let f be a nonnegative, integrable function on a measure space (X,M,?). Show that
?{x : f(x)=∞} = 0
and
?{x : f(x)>0} is σ-finite (that is, a countable union of sets of finite measure).