Answer TWO questions

• there is a function f such that t_n\nearrow f almost everywhere on I; and
• the sequence áò_I t_n ñ converges.

Show that, for any step function t such that t(x) £ f(x) almost everywhere on I,

(b) Define the set of upper functions on I and the integral of an upper function. Show that this integral is well-defined.

(c) Explain why it is desirable to extend the set of upper functions on I to the set L¹(I) of Lebesgue integrable functions on I. Give the definition of the space L¹(I) and of the integral ò_I f of an f Î L¹(I) and show that this integral is well-defined.
[Basic properties of integrals of upper functions may be assumed.]

2. (a) Let S Í \mathbbR. Define what is meant by saying that S has measure zero. Prove that if S is countable then it has measure zero.

(b) Write down Lebesgue's criterion for a function defined and bounded on an interval [a,b] to be Riemann integrable on [a,b].

Turn over

(c) Let f = 0 almost everywhere on \mathbbR. Show that f Î L¹ and òf = 0. Deduce that if f Î L¹ and g = f almost everywhere, then g Î L¹ and òf = òg.

(d) Let f be a non-negative element of L¹ and suppose òf = 0. Show that f = 0 almost everywhere. Deduce that if f ³ g, f,g Î L¹ and òf = òg then f = g almost everywhere.

3. (a) State carefully The Monotone Convergence Theorem and The Dominated Convergence Theorem for Lebesgue integrable functions.

(b) Let the functions g and f be defined for x > 0 by