Prelim Question help?
Prelim Question help?
http://www.math.northwestern.edu/gradua ... alf06.pdf
#5. It's kinda bustin' my balls here, I've done some stuff, but it doesn't seem to be going anywhere. Any thoughts/hints?
#5. It's kinda bustin' my balls here, I've done some stuff, but it doesn't seem to be going anywhere. Any thoughts/hints?

 Posts: 61
 Joined: Sun Apr 04, 2010 1:08 pm
Re: Prelim Question help?
So I assume this is probably referring to Lebesgue measure (it won't work for Borel measures).
Hints:
1. Reduce it to the case of the unit interval instead of the real line (easy)
2. If $$m(F) >0$$, $$1> \epsilon >0$$, then $$\exists I$$ such that $$\frac{m(I \cap F)}{m(I)} > 1\epsilon$$ and $$m(I) < \epsilon$$
Hints:
1. Reduce it to the case of the unit interval instead of the real line (easy)
2. If $$m(F) >0$$, $$1> \epsilon >0$$, then $$\exists I$$ such that $$\frac{m(I \cap F)}{m(I)} > 1\epsilon$$ and $$m(I) < \epsilon$$

 Posts: 61
 Joined: Sun Apr 04, 2010 1:08 pm
Re: Prelim Question help?
Btw, the $$I$$ above is an open interval (or closed if you want)
Re: Prelim Question help?
hey blitzer, are you sure your 2. is true? For example, look at the fat cantor set C. It has positive measure but m(I cap C)/m(I) is bounded above by some constant strictly less than 1.

 Posts: 61
 Joined: Sun Apr 04, 2010 1:08 pm
Re: Prelim Question help?
I'm 95% sure it's true. In fact, it's an exercise in Folland (number 30 in chapter 1). Specifically :
" If $$E \in \mathcal{L}$$ and $$m(E) > 0$$, for any $$\alpha < 1$$ there is an open interval $$I$$ such that $$m(E \cap I) > \alpha m(I)$$. "
I know it isn't true for the unit interval in your example. However, I think there does exist an interval by the outer regularity of Borel measures.
" If $$E \in \mathcal{L}$$ and $$m(E) > 0$$, for any $$\alpha < 1$$ there is an open interval $$I$$ such that $$m(E \cap I) > \alpha m(I)$$. "
I know it isn't true for the unit interval in your example. However, I think there does exist an interval by the outer regularity of Borel measures.
Re: Prelim Question help?
Okay, I see, you're right. I thought my statement on the fat cantor set held for every open interval I, but looks like my intuition was wrong.
BTW this result has a name: http://en.wikipedia.org/wiki/Lebesgue's_density_theorem
BTW this result has a name: http://en.wikipedia.org/wiki/Lebesgue's_density_theorem
Re: Prelim Question help?
Doing your part a) is easy, but I don't see how the problems follows from b). A little more help on the approximation?

 Posts: 61
 Joined: Sun Apr 04, 2010 1:08 pm
Re: Prelim Question help?
I'll try to do a sort of picture proof so I can leave out the annoying details. Let epsilon be greater than 0. Then by above (b) you can find an interval I such that F is pretty dense in the interval and that the measure of I is less than epsilon. Then you can cover the unit interval by translates using density of your set a_n:
[( {I \cap F} + a_{n_1} )( {I \cap F} + a_{n_2} ) ( {I \cap F} + a_{n_3}  ... ( {I \cap F} + a_{n_m} )  ]
The first  gap should be less than epsilon, the second  gap should be less than epsilon/2 and so on. You keep going until you can't fit any more so that the last  gap is less than epsilon. The translates of I are disjoint and those add up to at least 13*epsilon. When you intersect with F, you get measure at least
(13*epsilon)*(1epsilon).
Obviously if you add more translates of F you get only an increase of measure. Now since epsilon was arbitrary, the measure of G \cap unit interval = 1
[( {I \cap F} + a_{n_1} )( {I \cap F} + a_{n_2} ) ( {I \cap F} + a_{n_3}  ... ( {I \cap F} + a_{n_m} )  ]
The first  gap should be less than epsilon, the second  gap should be less than epsilon/2 and so on. You keep going until you can't fit any more so that the last  gap is less than epsilon. The translates of I are disjoint and those add up to at least 13*epsilon. When you intersect with F, you get measure at least
(13*epsilon)*(1epsilon).
Obviously if you add more translates of F you get only an increase of measure. Now since epsilon was arbitrary, the measure of G \cap unit interval = 1