I have decided to make the first test open-book, but only copies of Adams
and Guillemin are permitted, no papers or other writings.
the unit square in
is Lebesgue measurable and
- Without using the equality of Riemann and Lebesgue integrals show
- Give, with proofs, an example of a non-negative Lesbesgue measurable
function on which has finite Lebesgue integral but is not bounded.
- Explain why the formula
defines a countably additive finite measure on the -ring,
of all Lebesgue measurable subsets of
Show that completion
with respect to this measure is just
be a measure space and suppose that
is an increasing (i.e. pointwise
non-decreasing) sequence of simple measurable functions with
Show that the set consisting of those points
has -measure zero.
Richard B. Melrose