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Measure defined on Borel sets that it is finite on compact sets

  • Thread starter mahler1
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  • #1
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The problem statement

Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that:

a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open.

b)##\mu(E)=sup\{\mu(K), K \subset E\}##, where ##K## is compact.

Prove the following:

i. The open and compact sets belong to ##\mathcal H##.

ii. If ##\mu## is finite, ##\mathcal H## is a ##\sigma-##algebra.

iii. ##\mathcal H## coincides with the ##\sigma-##algebra of Borel.

The attempt at a solution

For i., maybe I could find an increasing sequence of compact sets ##\{K_n\}_{n \in \mathbb N}##(##K_n \subset K_{n+1}##) such that all are contained in ##E##, the problem is that I don't know how to construct this sequence; I suppose that, in an analogous way, I can construct a decreasing sequence of open sets ##\{G_n\}_{n \in \mathbb N}## (##G_{n+1} \subset G_n##) such that ##E## is contained in all of them.

For ii., it's easy to verify that ##\emptyset \in \mathcal H##, it remains to prove that if ##E \in \mathcal H##, then ##E^c \in \mathcal H##, and that if ##E_n \in \mathcal H## for a sequence of sets, then ##\bigcup_{n \in \mathbb N} E_n \in \mathcal H##. I couldn't prove that the complement of ##E## must be in ##\mathcal H##, I'll write what I did for countable unions:

Suppose ##E_n \in \mathcal H## for a sequence of sets, call ##E=\bigcup_{n \in \mathbb N} E_n##. By hypothesis, we have that ##\mu(E_n)## is finite for each ##n##. Given ##\epsilon>0##, we can choose for each ##n##, an open set ##G_n## : ##\mu(G_n)\leq \mu(E_n)+\dfrac{\epsilon}{2^n}##, if I call ##G=\bigcup_{n \in \mathbb N} G_n##, then ##E \subset G## and ##\mu(G) \leq \mu(E)+ \epsilon##. This means that ##\mu(E)## satisfies a). Analogously, we can show that ##\mu(E)## satisfies b), from here it follows ##E \in \mathcal H##.

For iii., assuming I could prove i., I can say that ##B \subset \mathcal H## since the open sets are contained in ##\mathcal H##, it remains to show that ##\mathcal H \subset B##.

I am pretty stuck in all three items, I would appreciate some help with this exercise and if someone could tell me if what I did for countable unions in ii. is correct.
 

Answers and Replies

  • #2
22,097
3,278
The problem statement

Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that:

a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open.

b)##\mu(E)=sup\{\mu(K), K \subset E\}##, where ##K## is compact.

Prove the following:

i. The open and compact sets belong to ##\mathcal H##.

ii. If ##\mu## is finite, ##\mathcal H## is a ##\sigma-##algebra.

iii. ##\mathcal H## coincides with the ##\sigma-##algebra of Borel.

The attempt at a solution

For i., maybe I could find an increasing sequence of compact sets ##\{K_n\}_{n \in \mathbb N}##(##K_n \subset K_{n+1}##) such that all are contained in ##E##, the problem is that I don't know how to construct this sequence; I suppose that, in an analogous way, I can construct a decreasing sequence of open sets ##\{G_n\}_{n \in \mathbb N}## (##G_{n+1} \subset G_n##) such that ##E## is contained in all of them.
You need to use that ##E## is actually compact (or open). So take ##E## compact. Then (b) shouldn't be too difficult. For ##A##, perhaps you should think of

[tex]\{x\in \mathbb{R}^n~\vert~d(x,E)<1/n\}[/tex]

For ii., it's easy to verify that ##\emptyset \in \mathcal H##, it remains to prove that if ##E \in \mathcal H##, then ##E^c \in \mathcal H##, and that if ##E_n \in \mathcal H## for a sequence of sets, then ##\bigcup_{n \in \mathbb N} E_n \in \mathcal H##. I couldn't prove that the complement of ##E## must be in ##\mathcal H##,
Take an ##E\in \mathcal{H}##. Let's prove ##(a)## for ##E^c##. You must find an open set ##G## such that ##E^c\subseteq G## and such that ##\mu(G) - \varepsilon<\mu(E^c)##. Take complements, then you need to find a certain closed subset of ##E##. Use that ##E\in \mathcal{H}## to find this.

I'll write what I did for countable unions:

Suppose ##E_n \in \mathcal H## for a sequence of sets, call ##E=\bigcup_{n \in \mathbb N} E_n##. By hypothesis, we have that ##\mu(E_n)## is finite for each ##n##.
I don't see why ##\mu(E_n)## is finite.

Given ##\epsilon>0##, we can choose for each ##n##, an open set ##G_n## : ##\mu(G_n)\leq \mu(E_n)+\dfrac{\epsilon}{2^n}##, if I call ##G=\bigcup_{n \in \mathbb N} G_n##, then ##E \subset G## and ##\mu(G) \leq \mu(E)+ \epsilon##.
Not sure if this is easy to see. We don't have necessarily that the ##G_n## and ##E_n## are pairswise disjoint. So you might not be able to use ##\sigma##-additivity.

For iii., assuming I could prove i., I can say that ##B \subset \mathcal H## since the open sets are contained in ##\mathcal H##, it remains to show that ##\mathcal H \subset B##.
But ##\mu## is a Borel measure, so it's only defined on the Borel sets.
 
  • #3
Dick
Science Advisor
Homework Helper
26,258
618
The problem statement

Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that:

a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open.

b)##\mu(E)=sup\{\mu(K), K \subset E\}##, where ##K## is compact.

Prove the following:

i. The open and compact sets belong to ##\mathcal H##.

ii. If ##\mu## is finite, ##\mathcal H## is a ##\sigma-##algebra.

iii. ##\mathcal H## coincides with the ##\sigma-##algebra of Borel.

The attempt at a solution

For i., maybe I could find an increasing sequence of compact sets ##\{K_n\}_{n \in \mathbb N}##(##K_n \subset K_{n+1}##) such that all are contained in ##E##, the problem is that I don't know how to construct this sequence; I suppose that, in an analogous way, I can construct a decreasing sequence of open sets ##\{G_n\}_{n \in \mathbb N}## (##G_{n+1} \subset G_n##) such that ##E## is contained in all of them.

For ii., it's easy to verify that ##\emptyset \in \mathcal H##, it remains to prove that if ##E \in \mathcal H##, then ##E^c \in \mathcal H##, and that if ##E_n \in \mathcal H## for a sequence of sets, then ##\bigcup_{n \in \mathbb N} E_n \in \mathcal H##. I couldn't prove that the complement of ##E## must be in ##\mathcal H##, I'll write what I did for countable unions:

Suppose ##E_n \in \mathcal H## for a sequence of sets, call ##E=\bigcup_{n \in \mathbb N} E_n##. By hypothesis, we have that ##\mu(E_n)## is finite for each ##n##. Given ##\epsilon>0##, we can choose for each ##n##, an open set ##G_n## : ##\mu(G_n)\leq \mu(E_n)+\dfrac{\epsilon}{2^n}##, if I call ##G=\bigcup_{n \in \mathbb N} G_n##, then ##E \subset G## and ##\mu(G) \leq \mu(E)+ \epsilon##. This means that ##\mu(E)## satisfies a). Analogously, we can show that ##\mu(E)## satisfies b), from here it follows ##E \in \mathcal H##.

For iii., assuming I could prove i., I can say that ##B \subset \mathcal H## since the open sets are contained in ##\mathcal H##, it remains to show that ##\mathcal H \subset B##.

I am pretty stuck in all three items, I would appreciate some help with this exercise and if someone could tell me if what I did for countable unions in ii. is correct.
I think you are trying to go too fast here. Let's just stick with i) for a while. You are dealing with ##\mathbb R^n##. You probably know a lot about ##\mathbb R^n##. You probably know Heine-Borel. Use that to attack i).
 

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