I have decided to make the first test open-book, but only copies of Adams and Guillemin are permitted, no papers or other writings.

- Set
the unit square in
and
suppose that
is Lebesgue measurable and
such that

Show that - Without using the equality of Riemann and Lebesgue integrals show
that

and

- 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 of with respect to this measure is just again. - Let 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 where has -measure zero.

Richard B. Melrose 2004-05-24