Prelim Question help?

Forum for the GRE subject test in mathematics.
Rikimaru
Posts: 25
Joined: Thu Feb 07, 2013 3:11 pm

Prelim Question help?

Postby Rikimaru » Mon May 27, 2013 3:47 pm

http://www.math.northwestern.edu/gradua ... al-f06.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?

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: Prelim Question help?

Postby blitzer6266 » Fri May 31, 2013 2:22 am

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

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: Prelim Question help?

Postby blitzer6266 » Fri May 31, 2013 2:25 am

Btw, the I above is an open interval (or closed if you want)

frgf
Posts: 24
Joined: Wed Jan 30, 2013 2:24 am

Re: Prelim Question help?

Postby frgf » Fri May 31, 2013 5:38 am

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.

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: Prelim Question help?

Postby blitzer6266 » Fri May 31, 2013 11:58 am

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.

frgf
Posts: 24
Joined: Wed Jan 30, 2013 2:24 am

Re: Prelim Question help?

Postby frgf » Fri May 31, 2013 9:43 pm

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

Rikimaru
Posts: 25
Joined: Thu Feb 07, 2013 3:11 pm

Re: Prelim Question help?

Postby Rikimaru » Mon Jun 03, 2013 9:48 am

Doing your part a) is easy, but I don't see how the problems follows from b). A little more help on the approximation?

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: Prelim Question help?

Postby blitzer6266 » Mon Jun 03, 2013 6:35 pm

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 1-3*epsilon. When you intersect with F, you get measure at least

(1-3*epsilon)*(1-epsilon).

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




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