Last updates: 5 March 2011
Does there exist an infinite -algebra which has only countably many members?
- Prove an analogue of [Ru, Theorem 1.8] for functions.
- Prove that if is a real function on a measurable space
is measurable for every rational , then is
- Let and
be sequences in
and prove that
- (a) .
- (b) ,
provided none of the sums is of the form .
Show by an example that strict inequality can hold.
If for all then
are measurable. Prove that the sets
Prove that the set of points at which a sequence of measurable real-valued functions converges
(to a finite limit) is measurable.
Let be an uncountable set, let be the collection
of all sets such that either
or is at most countable,
and define in the first case,
in the second.
Prove that is a -algebra on
and that is a measure on . Describe the
corresponding measurable functions and their integrals.
is measurable for ,
for every , and
Prove that then
and show that this conclusion does not follow if the condition
"" is omitted.
if is odd, and
if is even. What is the relevance of this example to Fatou's lemma?
- Suppose is a positive measure on ,
is measurable, and
and is a constant. Prove that
Hint: If , the integrands are dominated
by . If , Fatou's
lemma can be applied.
is a sequence of bounded complex measurable functions
on , and
uniformly on . Prove that
- Show that
in [Ru, Theorem 1.41], and hence prove the theorem without any reference to integration.
- Suppose . Prove that to each there exists
a such that
- Show that [Ru, Proposition 1.24(c)] is also true when .
Notes and References
These exercises are taken from [Ru] for a course in "Measure Theory" at the Masters level at University of Melbourne.
Real and complex analysis, Third edition, McGraw-Hill, 1987.