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

[mahler1]

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.

 

[micromass]

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

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

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.

 

[Dick]

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).