Math Problems: Integration

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Problem 1

1. If f: [0,1] -> R is continuous and such that for all x in [0,1], int(f(t)dt,[x,1]) >= (1-x^2)/3, then prove: int(f^2(t)dt,[0,1]) >= 1/9.
Solution

Problem 2

2. Given f: [0,1] -> R absolutely continuous such that f(0)=0 and int(|f '(x)|^2,[0,1]) < +oo, find lim(x^(-f(x)/2),x->0+).
Solution

Problem 3

3. Show that B_m x B_n = B_(m+n), where B_k is the Borel sigma-algebra of R^k.
Solution

Problem 4

4. Show that L_k, the Lebesgue sigma-algebra of R^k, is the completion of B^k.
Solution

Problem 5

5. Show that L_m x L_n <> L_(m+n), where L_k is the Lebesgue sigma-algebra on R^k.
Solution

Problem 6

6. If a function f: [a,b] -> R is absolutely continuous on ]c,b] for every c > a and is continuous with bounded variations on [a,b], show that f is absolutely continuous on [a,b].
Solution

Problem 7

7. (a) Show that the image of a null set E by an absolutely continuous function f: [a,b] -> R is null; (b) prove that the image of a Lebesgue-measurable set E by f is Lebesgue-measurable.
Solution

Problem 8

8. Let g: [a,b] -> R be a nondecreasing absolutely continuous function and f: [g(a),g(b)] -> R be a continuous function. Show that int(f(t)dlambda(t),[g(a),g(b)]) = int(f(g(t))g'(t)dlambda(t),[a,b]). Hint: Consider the function F defined by F(x)=int(f(t) d\lambda(t),g(a)..g(x)). (lambda is the Lebesgue measure).
Solution

Problem 9

9. Let f be a real valued Lebesgue-measurable function on R^k. Prove that there exist two Borel functions g and h such that g(x)=h(x) a.e [m] and for all x in R^k, g(x)<= f(x) <= h(x).
Solution

Problem 10