Answer TWO questions

(ii) Let áI_n ñ be a sequence of open intervals in I = ]0,1[ which covers all the rational points in I and such that the sum of the lengths of these intervals is less than 1/2. Let T = ÈI_n and let c_T be the characteristic function of T, that is, c_T(x) = 1 if x Î T and c_T(x) = 0 if x\not Î T. Show that c_T is in L^inc(I) but -c_T is not in L^inc(I).

(iii) What does the above example say about the space L^inc(I)? Explain very briefly how this difficulty is remedied in the definition of L¹(I).

2. (i) State carefully Levi's Monotone Convergence Theorem and Lebesgue's Dominated Convergence Theorem for Lebesgue integrable functions.

(ii) Give an example of a sequence of functions áf_nñ such that each f_n is integrable, the sequence converges to an integrable function f, but the sequence of integrals áòf_nñ does not converge to òf. Explain why this sequence does not contradict Levi's or Lebesgue's theorems.

(iii) Show that if f is a non-negative function in L¹ and the Lebesgue integral òf = 0 then f = 0 a.e.

(iv) Give an example of a sequence of functions áf_nñ which is bounded, each f_n is Riemann integrable, and the sequence converges to a limit function f which is Lebesgue integrable but not Riemann integrable. Explain clearly why your choice of sequence satisfies all these conditions.

(v) Let the function f be defined by f(x) = 1/(1+x²) for all x ³ 0. By finding a sequence of functions which converges appropriately to f show that the Lebesgue integral ò₀^¥ f exists and find the value of this integral.

3. (i) Let I be the interval [a,b]. Suppose that for any function f:I® \mathbbR continuous on I = [a,b] and for any sub-interval J of I there is defined a number denoted by S_J(f) satisfying the following properties:

• If J₁ = [p,q], J₂ = [q,r] and J₃ = [p,r] are three sub-intervals of I then

  SJ3(f) = SJ1(f)+SJ2(f);

• If m £ f(x) £ M for all x in the sub-interval J = [p,q] then

  m(p-q) £ SJ(f) £ M(p-q).

Let F:I®\mathbbR be defined by F(x) = S_Ix(f), where I_x is the interval [a,x]. Show that F¢(x) exists and is equal to f(x) for all x Î [a,b].

Also, if H:[a,b]® \mathbbR is any other function such that H¢(x) = f(x) for all x Î [a,b], show that S_[a,b](f) = H(b)-H(a).

(ii) Now, in the above, let a = 1 and f(t) = 1/t for all t ³ 1, and let g(x) = 4xF(x). By differentiating x²(2F(x)-1) find