Help with Measure Theory: Sup & Inf of B_n

[Mathman23]

If E is a non empty set and$$(B_n)_{n \geq 1}$$ are elements in the set $$2^E$$.

I then need help showing the following:

$$lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcup_{n\, =\, 1} ^{\infty}\, B_n$$

if and only if $$B_n\, \subseteq\, B_{n+1}$$, for all $$n\, \geq\, 1$$,

Also I need to show

$$lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcap_{n=1} ^{\infty} B_n$$

if and only if $$B_n\, \supseteq\, B_{n+1}$$, for all $$n\, \geq\, 1$$

I know that for every sequence $$(a_n)_{n\, \geq\, 1}$$ of elements in the set $$- \infty\ \union\ \mathbb{R}\ \union\ \infty$$.

$$lim_n\, sup\, a_n\, =\, inf(M_n|\, n\, \geq\, 1)$$, where $$M_n\, :=\, sup(a_k|\, k\, \geq\, n},\, n\, \geq\, 1$$.

$$lim_n\, inf\, a_n\, =\, inf(m_n|\, n\, \geq\, 1)$$, where $$m_n\, :=\, sup(a_k|\, k\, \geq\, n},\, n\, \geq\, 1$$.

But could somebody please give me a hint or an idear on how to use this fact to show the original task?

Sincerely Fred

 

[AKG]

This has nothing to do with measure theory, it's a purely set theoretic question. Start by writing down the definitions of lim sup and lim inf. And why am I seeing so many people write "idears" lately? The word is "idea".

 

[StatusX]

The limsup and liminf are defined whenever there is a partial order on a set. For real numbers, this is the normal "less than or equal to" order. For sets is its the "is a subset of" order. The sup of a collection of sets is defined just like it is for real numbers, ie, A is the sup of {B_n} iff 1) A contains every B_n and 2) If C contains every B_n, then C contains A. Can you figure out what A is here, and also in the inf case? Then use this in the definition of limsup and liminf.

 

[Mathman23]

Is A the sum of all intersecting partions that I mention ?

I mean $$= \bigcap \bigcup _{n=1} ^{\infty} B_n$$
??

I have a second question?

If E = R, where R being the set of all real numbers.

Where $$B_n = [0, x_n]$$ $$n \geq 1$$

is limited sequence of positive real numbers.

I need to show here

$$[0, lim , sup x_n [ \subseteq lim_{n} , sup B_n \subseteq [0, lim , sup x_n]$$

Any idears here ?

Sincerely Yours
Fred

 

[StatusX]

I don't know what A is supposed to be. And show what you've tried on the second question.

 

[Mathman23]

I don't know what A is supposed to be. And show what you've tried on the second question.

$$[0, lim \ sup x_n [ \subseteq lim_{n} sup B_n \subseteq [0, lim \ sup x_n]$$

should I treat $$x_{n} \subseteq B_{n}$$

and $$lim sup x_n = \bigcap_{j=1} \bigcup_{n=j} ^{\infty} x_n$$

if this is true then

$$[0, lim_{n} \ sup x_n [ \subseteq lim_{n} sup B_n \subseteq [0, lim_{n} \ sup x_n]$$

?

Sincerely Fred

 

[StatusX]

The x_n are numbers, and the B_n are sets, so you can't compare them like you're doing. Before you can do this problem, you need to have figured out what the limsup of a sequence of sets is. Have you done this?

 

[Mathman23]

What is what I have trouble with, do I treat x_n as a scalar in relation with B_n??

Sincerely Fred

 

[StatusX]

What do scalars have to do with anything? There aren't any vector spaces here. Don't overcomplicate things. Again, what is the limsup of a sequence of sets?

 

[Mathman23]

I recon that must be

$$lim_{n \rightarrow \infty} sup x_n = lim (sup x_m), m \geq n$$

??

Sincerely Fred

What do scalars have to do with anything? There aren't any vector spaces here. Don't overcomplicate things. Again, what is the limsup of a sequence of sets?

 

[StatusX]

You're not getting anywhere. Take some time and think about the problem. Post something once you've worked on it for a while.

 

[Mathman23]

I'm sorry, but could you please give me hint here?

/Fred


p.s. My father has very sick these last couple of weeks (heart trouble), and I have therefore not been able to get anywhere with this problem. So therefore I know its much to ask, but if you could be a kind soul and help me answer this problem, then I will never ask for anything this big again.

You're not getting anywhere. Take some time and think about the problem. Post something once you've worked on it for a while.

 

[StatusX]

So what do you want from me? All I can do is try to explain the concepts to you, and you're not taking the time to try and understand them. If you're looking for someone to do the problem for you, it's not going to happen. Ask specific, meaningful questions and I'll try to answer them as clearly as possible. But please take a little time to ask the best questions you can.