Existence of a strange measure

Question

The answer to this question must be known, but I do not know where to find it. It is related to the Ulam measures I believe.

Question. Is there a finitely additive measure defined on all subsets of positive integers $\mathbb{N}$, with values into $\{0,1\}$, (only two values) that is $$ \mu:2^{\mathbb{N}}\to \{0,1\} $$ such that $\mu(\{n\})=0$ for every $n\in\mathbb{N}$ and $\mu(\mathbb{N})=1$ ?

I believe such a result is needed in a proof of the co-area inequality that is stated in Coarea inequality, Eilenberg inequality. I am working with my student on some generalizations of that result so this is a research related question.

Answer 1

The answer is yes. I wrote a proof using YCor's comment.

Theorem. There a finitely additive measure defined on all subsets of positive integers $\mathbb{N}$, with values into $\{0,1\}$, (only two values) that is $$ \mu:2^{\mathbb{N}}\to \{0,1\} $$ such that $\mu(\{n\})=0$ for every $n\in\mathbb{N}$ and $\mu(\mathbb{N})=1$.

Proof. Let $$ \mathcal{F}_0=\{A\subset\mathbb{N}:\, \mathbb{N}\setminus A\ \ \text{is finite}.\} $$ $\mathcal{F}_0$ is a filter. By a filter we mean here a family $\mathcal{F}\subset 2^\mathbb{N}$ with the following properties

1. $\mathbb{N}\in \mathcal{F}$,
2. $\emptyset\not\in\mathcal{F}$,
3. $A\in\mathcal{F}$, $A\subset B$ $\Rightarrow$ $B\in\mathcal{F}$,
4. $A,B\in\mathcal{F}$ $\Rightarrow$ $A\cap B\in\mathcal{F}$.

Lemma. There is a family $\mathcal{F}\subset 2^{\mathbb{N}}$ such that $\mathcal{F}_0\subset\mathcal{F}$ and $\mathcal{F}$ has properties 1.-4. and also property

5. $A\subset\mathbb{N}$ $\Rightarrow$ $A\in\mathcal{F}$ or $\mathbb{N}\setminus A\in\mathcal{F}$.

Remark. A family $\mathcal{F}\subset 2^\mathbb{N}$ satisfying properties 1.-5. is called an ultrafilter.

Proof. Filters containing $\mathcal{F}_0$ are ordered by the inclusion. The union of a chain of filters is a filter (this is obvious). Therefore by the Kuratowski-Zorn lemma there is a maximal filter $\mathcal{F}$ that contains $\mathcal{F}_0$. It remains to show that $\mathcal{F}$ satisfies property 5. (it has properties 1.-4. since it is a filter).

Suppose to the contrary that there is $A\subset\mathbb{N}$ such that $$ (*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A\not\in\mathcal{F} \quad \text{and} \quad \mathbb{N}\setminus A\not\in \mathcal{F}. $$ Define $$ \widetilde{\mathcal{F}}= \{ E\subset \mathbb{N}:\, \exists B\in\mathcal{F}\ \ A\cap B\subset E\}. $$ Note that $\mathcal{F}\subset\widetilde{\mathcal F}$, because for $B\in\mathcal{F}$, $A\cap B\subset B$ so $E=B\in\widetilde{\mathcal{F}}$. Also $\mathcal{F}\subsetneq\widetilde{F}$ is a propert subset, because $\mathbb{N}\in\mathcal{F}$, $A\cap\mathbb{N}\subset A$ so $A\in\widetilde{\mathcal F}$ by definition and hence $A\in\widetilde{\mathcal F}\setminus\mathcal{F}$.

It remains to prove that $\widetilde{\mathcal F}$ is a filter (has properties 1.-4.) to reach a contradiction with the maximality of $\mathcal{F}$.

Clearly $\widetilde{\mathcal F}$ has properties 1., 3., and 4. (by property 4. for $\mathcal{F}$).

It remains to prove property 2. Suppose to the contrary that $\emptyset\in \widetilde{\mathcal F}$. Then $A\cap B=\emptyset$ for some $B\in\mathcal{F}$. Then $B\subset\mathbb{N}\setminus A$ so $\mathbb{N}\setminus A\in\mathcal{F}$ by property 3. for $\mathcal{F}$ which contradicts $(*)$. The proof is complete. $\Box$

Now we define the finitely additive measure $$ \mu:2^{\mathbb{N}}\to \{0,1\} $$ such that $\mu(\{n\})=0$ for every $n\in\mathbb{N}$ and $\mu(\mathbb{N})=1$ as follows $$ \mu(A)= \begin{cases} 1 & \text{if $A\in \mathcal{F}$},\\ 0 & \text{if $A\not\in\mathcal{F}$}. \end{cases} $$ Finite-additivity of $\mu$ follows from the fact that no two subsets with measure $1$ can be disjoint.

Note: Countable additivity fails because $$ \mathbb{N} = \bigcup_{n=1}^\infty \{n\} \quad \text{but} \quad \mu(\mathbb{N}) \neq \sum_{n=1}^\infty \mu(\{n\}) \ . $$

$\Box$

Answer 2

Just about any form of the axiom of choice can be used to prove this. I like topology, so here's a proof using Tychonoff's theorem.

Consider the space $2^{2^\mathbb{N}}$ of all functions $2^{\mathbb{N}}\to\{0,1\}$ under the product topology. By Tychonoff's theorem this is compact, and it is easy to check from the definition of the product topology that the set $\mathcal{F}$ of all finitely-additive $\{0,1\}$-valued measures on $\mathbb{N}$ is closed in $2^{2^\mathbb{N}}$. Then the sequence of measures $\mu_k\in\mathcal{F}$ defined by $$ \mu_k(S) = \begin{cases}1 & \text{if }k\in S,\\ 0 & \text{if }k\notin S\end{cases} $$ must have a limit point $\mu\in\mathcal{F}$. Then $\mu(\mathbb{N})=1$ since $\mu_k(\mathbb{N})=1$ for all $k$, and $\mu(\{n\}) = 0$ for all $n\in\mathbb{N}$ since $\mu_k(\{n\})=0$ whenever $k>n$.

Answer 3

My comment seems to be buried so I'd like to repeat it here. There is a simple C*-algebra construction that answers the question. The quotient space $l^\infty/c_0$ is a unital commutative C*-algebra, so there is a $*$-isomorphism $\Phi: l^\infty/c_0 \cong C(X)$ for some compact Hausdorff space $X$. For any $x \in X$, define $\mu_x(S) = \Phi(1_S)(x)$.

Every such $\mu_x$ works. In fact, this establishes a 1-1 correspondence between the set of non-principal ultrafilters on $\mathbb{N}$ and the spectrum of $l^\infty/c_0$.

Answer 4

This can be proved without introducing ultrafilters by name, by doing "finitary measure theory" and using Zorn's lemma.

An algebra $A$ on a set $X$ is just a $\sigma$-algebra without the $\sigma$, i.e. $\newcommand{\powerset}{\mathcal{P}}A \subseteq \powerset(X)$ and is closed under finite unions and complements (and therefore all other Boolean operations).

Let $\newcommand{\N}{\mathbb{N}}F \subseteq \powerset(\N)$ be the set of finite sets and their complements (so-called cofinite sets). This is an algebra. Furthermore, we can define a 2-valued finitely-additive measure $\mu : F \rightarrow \{0,1\}$ to be $0$ on the finite sets and $1$ on the cofinite sets. The existence of the required 2-valued finitely-additive measure on $\powerset(\N)$ then follows from:

Proposition For any algebra $A \subseteq \powerset(X)$ and finitely-additive 2-valued measure $\mu : A \rightarrow \{0,1\}$, there exists a finitely-additive measure $\overline{\mu} : \powerset(X) \rightarrow \{0,1\}$ extending $\mu$.

Proof: Most of the difficulty is in believing that it's true. We use Zorn's lemma. The poset consists of pairs $(B,\nu)$ where $B \supseteq A$ is an algebra of sets, and $\nu : B \rightarrow \{0,1\}$ is a finitely-additive measure extending $\mu$. The order relation $(B_1,\nu_1) \leq (B_2,\nu_2)$ is defined to hold when $B_1 \subseteq B_2$ and $\nu_2$ extends $\nu_1$. Every chain in this poset has an upper bound - we just take the union of algebras (this is the step that fails for $\sigma$-algebras) and define the measure on the union in the obvious way.

Let $(B,\nu)$ be a maximal element in the poset. Suppose for a contradiction that $B \neq \powerset(X)$, so there is some $U \in \powerset(X) \setminus B$. We contradict the maximality of $B$ by extending $\nu$ to a larger algebra $B'$ including $U$. Define $B' = \{ (U \cap S_1) \cup (\lnot U \cap S_2) \mid S_1, S_2 \in B \}$. It is clear that $B \subseteq B'$ and $U \in B'$, and with a little Boolean reasoning we can prove that for all $S_1,S_2,T_1,T_2 \in B$: $$ ((U \cap S_1) \cup (\lnot U \cap S_2)) \cup ((U \cap T_1) \cup (\lnot U \cap T_2))\\ = (U \cap (S_1 \cup T_1)) \cup (\lnot U \cap (S_2 \cup T_2)) $$ and $$ \lnot ((U \cap S_1) \cup (\lnot U \cap S_2)) = (U \cap \lnot S_1) \cup (\lnot U \cap \lnot S_2) $$ This proves that $B'$ is an algebra.

Now, define $d \in \{0,1\}$ to be the "outer measure" of $U$, i.e. $d = 0$ if there exists $S \in B$ such that $U \subseteq S$ and $\nu(S) = 0$, otherwise $d = 1$. Without loss of generality we can take $d = 1$, because we can exchange the roles of $U$ and $\lnot U$. We define $\nu'((U \cap S_1) \cup (\lnot U \cap S_2)) = \nu(S_1)$. This is well-defined because if $(U \cap S_1) \cup (\lnot U \cap S_2) = (U \cap T_1) \cup (\lnot U \cap S_2)$, then $U \cap S_1 = U \cap S_2$, so $U \subseteq \lnot (S_1 \triangle S_2)$, so as $d = 1$, $\nu(\lnot (S_1 \triangle S_2)) = 1$, and therefore $\nu(S_1) = \nu(S_2)$. The identities we used to prove that $B'$ is an algebra can then be used to prove that $\nu'$ is finitely additive, and it follows directly from the definition that it extends $\nu$. So we successfully contradicted the maximality of $B$. $\square$

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Of course, I actually think ultrafilters are a good thing to know about, both in topology and logic. There is also no metamathematical benefit in doing it this way - over ZF the existence of a non-principal ultrafilter on $\N$ and a finitely-additive 2-valued measure on $\powerset(\N)$ are equivalent. The above proof is based on something I came up with while reproving Stone duality in the case where the points of the Stone space are defined to be Boolean homomorphisms into $2$, rather than ultrafilters. The proposition above is a special case of the fact that complete Boolean algebras (such as $2$) are injective objects in the category of Boolean algebras.