Last updates: 8 March 2011
Measure Theory Problem Set 2
Let be a measure space.
be measurable. Let . Show that the set
is in .
Let be a measure space
and let be a measure.
For a simple function, give
the definition of .
Let be integrable. Give the
definition of .
Working directly from the definition of the integral show that if
is defined by
- Let be a compact subset of . Let
be continuous. Let
. Show that there exist
such that for all ,
- Show that if are Borel probability measures
on a standard Borel space then we can find (positive)
- Let be the outer measure used to
define Lebesgue measure . Carefully define
and Lebesgue measurable sets.
Show that if is Lebesgue measurable
then there are Borel with
- Let be two σ-algebras on a
set . Let be a σ-finite measure on
with respect to . Show that there is a function
which is measurable
with respect to such that on any
- Let be a standard Borel space and let be a finite
atomless Borel measure on . Then for all
we can find with .
- Show that there exists a subset which is not Lebesgue measurable.
- Let be a compact metric space. Let be
Borel. Let be finite Borel measures on with the property that
for all Borel
for all Borel
Let be the signed
(i.e. ). Show that
equals the supremum of
- Let , the unit
interval in its usual topology. Let be
the space of probability measures on equipped with the topology generated by
the basic open sets
continuous functions from to
. Let be the open interval . Is
closed in this topology?
- Let be a finite measure space. Let
be a sequence of measurable functions which converge pointwise to . Then for any
and converging uniformly to
- Equip with the discrete topology and
be the collection of all functions equipped with the product topology. For
a sequence of distinct elements of
a sequence of elements of
Let be the collection of all finite unions of the sets of the form
. Show that
extends to a function on
which is σ-additive on its domain.
Notes and References
These exercises are taken from problems by G. Hjorth for a course in "Measure Theory" at the Masters level at University of Melbourne.
Real and complex analysis, Third edition, McGraw-Hill, 1987.