1. ## Some measure Theory

1. Let $\displaystyle S = \{x \in[0,1]:$no 4s occur in any decimal expansion of$\displaystyle x\}$. Find $\displaystyle m^{*}(S)=0$. I think the answer is 0, but have no idea how....

2. Given a sequence of subsets of $\displaystyle \mathbb{R}$, say $\displaystyle \{A_i\}_{i=1}^{\infty}$ give an example to show that lim sup A_n and lim inf A_n are different.

3. I don't understand South Park's argument below. Could anyone please explain? (the <---- implication)

thanks

Originally Posted by southprkfan1
Question: $\displaystyle A_{\epsilon} \leq E \leq B_{\epsilon}$ makes no sense since they are sets. Do you mean they are subsets or their measures are less than or equal? It looks like you mean subsets, so I'll use that, but the proof is nearly the same either way.

-->Suppose E is Measurable

If E is measurable, then there exists a G-delta set, G and a F-sigma set, F such that $\displaystyle F \subseteq E \subseteq G$ and $\displaystyle m(G \setminus F) = 0$. Thus, let $\displaystyle A_{\epsilon} = F$ and $\displaystyle B_{\epsilon} = G$

<--Suppose for any there are lebesgue measurable sets such that and that $\displaystyle m(B_{\epsilon} \setminus A_{\epsilon})<\epsilon$.

Let $\displaystyle A = \bigcap_{n=1}^{\infty}A_n$ and $\displaystyle B = \bigcap_{n=1}^{\infty}B_n$

Clearly, $\displaystyle A \subseteq E \subseteq B$ and A and B are both measurable.

And $\displaystyle m(B \setminus A) \leq 1/n$ for all n; thus, $\displaystyle m(B \setminus A) = 0$.

So E differs by A (or B) by a zero set, so E is measurable.

2. Originally Posted by davidmccormick
1. Let $\displaystyle S = \{x \in[0,1]:$no 4s occur in any decimal expansion of$\displaystyle x\}$. Find $\displaystyle m^{*}(S)=0$. I think the answer is 0, but have no idea how....

2. Given a sequence of subsets of $\displaystyle \mathbb{R}$, say $\displaystyle \{A_i\}_{i=1}^{\infty}$ give an example to show that lim sup A_n and lim inf A_n are different.

3. I don't understand South Park's argument below. Could anyone please explain? (the <---- implication)

thanks
1. I'll try to get back.

2. Let $\displaystyle A_n$ = {1} if n is even and {-1} if n is odd, then limsup = 1, liminf = -1

3. What exactly are you having trouble with? I admit I skipped a few steps at the end, and the 1/n should just be an epsilon, if that helps.

3. Well what are the sets, $\displaystyle A$ and $\displaystyle B$? Are they just the $\displaystyle A_{\epsilon}$ and $\displaystyle B_{\epsilon}$ given in the question? because then the next 2 lines are just rewriting the statement of the problem.....

4. Originally Posted by davidmccormick
Well what are the sets, $\displaystyle A$ and $\displaystyle B$? Are they just the $\displaystyle A_{\epsilon}$ and $\displaystyle B_{\epsilon}$ given in the question? because then the next 2 lines are just rewriting the statement of the problem.....
Ok I see the problem now, it should read

$\displaystyle A = \bigcap_{n=1}^{\infty}A_{1/n}$ $\displaystyle B = \bigcap_{n=1}^{\infty}A_{1/n}$

With the rest as it was before.