Constructing an outer measure on a set whose measurable sets are exactly a given sigma algebra on the set.

Consider an arbitrary set $X$ and an arbitrary $\sigma$-algebra $\mathcal{M}$ on $X$.

My question is that can one construct an outer measure on the set $X$ whose measurable sets is exactly the collection $\mathcal{M}$.

I tried to find an answer for finite sets and found this proposition to be true.

The solution is, let $X$-finite set, $\mathcal{M}$-algebra on $X$(and hence a $\sigma$-algebra on $X$) and $\mu_{0}(A)=|A|$ (cardinality of A) $\forall A \in \mathcal{M}$. It is easy to verify that $\mu_{0}$ is a pre-measure on the algebra $\mathcal{M}$ as $\mu_{0}(\emptyset)=0$ and it is countably additive (over here only finite additivity suffices).

Thus we construct the outer measure on $\mathcal{P}(X)$ using $\mu_{0}$, call it $\mu^*$.

Let $B\subset X(\notin \mathcal{M})$. So $(X\setminus B) \notin \mathcal{M}$. Then $\mu^*(B)>|B|$ as all the elements of $\mathcal{M}$ containing $B$ has higher cardinality.

If possible $B$ is $\mu^*$-measurable.

So we can check $|X|=\mu^*(X)= \mu^*(B) + \mu^*(X\setminus B)>|B|+|X\setminus B|=|X|$. Hence this is a contradiction and hence $B$ is not $\mu^*$-measurable.

Hence the only $\mu^*$-measurable sets are the sets in $\mathcal{M}$.

I have no idea how to proceed with this problem for infinite sets and maybe more general cases. Any kind of help and idea is highly appreciated.

Thanks.

Edit: I also figured out that even in infinite sets, if the concerned $\sigma$- algebra is a finite one, we can define a pre-measure on it in the same way and check that these are the only measurable sets.

Solution 1:

Second attempt for a counter-example. :-)

Let $X=X_0\times\{1,2\}$ with some uncountable set $X_0$, and let $\mathcal{M}=\{A\times\{1,2\}:~A\subset X_0,~\text{either $A$ or $X_0\setminus A$ is countable.}\}$

Suppose that there is an outer measure $\mu^*$ on $P(X)$ such that $\mathcal{M}$ is precisely the set of $\mu^*$-measurable sets.

Notice that for every $x\in X_0$, the sets $\{(x,1)\},\{(x,2)\}\notin\mathcal{M}$, so these sets are not measurable; therefore $\mu^*\big(\{(x,1)\}\big)>0$, $\mu^*\big(\{(x,2)\}\big)>0$ and $\mu^*\big(\{(x,1),(x,2)\}\big)>0$.

Take some $A_0\subset X_0$ such that both $A_0$ and $X_0\setminus A_0$ are uncountable, and let $A=A_0\times\{1,2\}$. Obviously $A\notin\mathcal{M}$, so $A$ is not measurable. Hence, there is some $B\subset X$ such that $\mu^*(B\cap A)+\mu^*(B\setminus A)>\mu^*(B)$. It follows that $m:=\mu^*(B)$ is finite.

Let $B_0=\{x\in X_0:\text{ $(x,1)\in B$ or $(x,2)\in B$}\}$ be the projection of $B$ on $X_0$. We shall prove that $B_0$ is countable.

Take a positive integer $k$ and arbitrary elements $c_1,\ldots,c_n\in B_0$ such that $\mu^*\big(B\cap\{(c_i,1),(c_i,2)\}\big)>\frac1k$. Let $C=\{c_1,\ldots,c_n\}\times\{1,2\}$; since $\{(c_i,1),(c_i,2)\}$ is measurable, we get $$ m=\mu^*(B) \ge \mu^*(B\cap C)=\sum_{i=1}^n \mu^*\big(B\cap\{(c_i,1),(c_i,2)\}\big)>\frac{n}{k}, $$ so $n<km$. Hence, there are only finitely many elements $c\in B_0$ such that $\mu^*\big(B\cap\{(c_i,1),(c_i,2)\}\big)>\frac1k$.

Since $\mu^*\big(B\cap\{(c,1),(c,2)\}\big)>0$ for all $c\in B_0$, this proves that $B_0$ is countable.

Now we can replace $A$ by $A'=A\cap(B_0\times\{1,2\})\in\mathcal{M}$. Note that $B \cap A = B \cap A'$, so that we get a contradiction by $$ \mu^*(B) < \mu^\ast(B\cap A)+\mu^*(B\setminus A) = \mu^\ast (B\cap A')+\mu^*(B\setminus A') = \mu^*(B). $$

Solution 2:

Under certain additional assumptions it is possible to construct an outer measure s.t. the collection of $\mu^*$-measurable sets coincides with $\mathcal{M}$ (compare your example with the following result). Suppose that $(X,\mathcal{M})$ is endowed with a measure $\mu$ and consider the usual construction of the corresponding outer measure $\mu^*$, i.e. for $A\subset X$, \begin{align} \mu^*(A):\!&=\inf\!\left\{\sum_j \mu(B_j):A\subset \bigcup_j B_j, \{B_j\}\subset\mathcal{M}\right\} \\ &=\inf\{\mu(B):A\subset B\in\mathcal{M}\}. \end{align}

If $(X,\mathcal{M},\mu)$ is complete and $\mu$ is $\sigma$-finite, then $\mathcal{M}$ coinsides with $\mathcal{M}^*$.

Proof. Suppose that $\mu(X)<\infty$. For $A\in \mathcal{M}^*$, there exists $B\in \mathcal{M}$ s.t. $A\subset B$ and $\mu^*(A)=\mu(B)^{(1)}$. Thus, $\mu^*(B\setminus A)=0$ and (using the same argument) there exists $C\in \mathcal{M}$ s.t. $B\setminus A \subset C$ and $\mu(C)=\mu^*(B\setminus A)=0$. Since $\mu$ is complete, $A=B\setminus(B\setminus A)\in \mathcal{M}$. As for the general case write $X=\bigcup_j X_j$ with $\mu(X_j)<\infty$. Then, using the above argument, $\mathcal{M}^*\ni A=\bigcup_j (A\cap X_j)\in\mathcal{M}$.

This shows that $\mathcal{M}^{*}\subseteq\mathcal{M}$. The other inclusion is obvious. $\square$

${}^{(1)}$ By the definition of $\mu^*$ we can find a decreasing family of sets $\{B_n\}\subset \mathcal{M}$ s.t. $A\subset B_n$ for each $n$ and $\mu^*(A)=\mu^*(B)=\mu(B)$, where $B\equiv\bigcap_{n\ge 1}B_n\in\mathcal{M}$.