I need help with my real analysis homework assignments questions 1,2 and 3. I've attached both the assignment and the "book" we use in the files.

In the next two problems we will calculate some integrals. Don't forget to explain how you go from Lebesgue integrals to Riemann integrals and back all the time.

For this you have to use Example 5.16 and Theorem 6.12 or sometimes you have to repeat the argument given there.

Problem 2. Let denote the Lebesgue measure on ( R;B (R )). Let f:R ! Rbe given by f(x ) = 1 p x x 2 (0;1]; 0 ; otherwise.

(a)Show that fis measurable via Proposition 4.5(iii).

(b)Show that fis integrable via Example 5.16 and a limit procedure.

(c)Let ( q n ) n 1 be an enumeration of Q. Let g:R ! Rbe given by g(x ) = 1 X n =1 2 n f (x q n ). Show that gis integrable. 1 Hint: Use one of the three convergence theorems or their consequences. Also use Exercise 5.10.

Problem 3. Let f: (0 ;1 )! Rbe given by f(x ) = sin( x) e x 1 .

(a)Show that fis integrable with respect to the Lebesgue measure on (0 ;1 ).

Hint: Use Exercise 5.7(a) and sin( x) x(for example). It could help to split (0 ;1 ) = (0 ;1) [[1;1 ).

(b)Show that for every k 1, Z (0 ;1 )e kx sin( x) d (x ) = 1 1 + k2 .

Hint: You may use that a primitive of e kx sin( x) is given by e kx (k sin( x) + cos( x)) k 2 + 1 .

(c)Use the dominated convergence theorem to show that Z (0 ;1 )f d = 1 X k =1 1 k 2 + 1 :

Hint: Use a geometric series to rewrite 1 e x 1 = e x 1 e x .

DEADLINE: January 10th at 23.00. Hand in electronically via brightspace.

GOOD LUCK!

Problem 1: 3p, Problem 2: 3p, Problem 3: 3p.

The total grade is the sum of the points +1. 1 The function is unbounded around any rational point of R. Making a plot this function would be impossible 1