Properties of measures that are not countably additive but have countably additive null ideals

Question

This is a very naive question, maybe more of a reference request than anything else.

Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has a countably additive null ideal if $m(\cup_{n=1}^\infty A_n) = 0$ whenever $A_n \in \mathcal X$ and $m(A_n)=0$ for all $n$.

Of course if $m$ is a countably additive measure, then $m$ has a countably additive null ideal.

If $m$ is a merely finitely additive probability measure (i.e. finitely but not countably additive and such that $m(X)=1$) it may or may not have a countably additive null ideal. In a typical example of a merely finitely probability, the null ideal is not countably additive: extend the natural density function to a probability measure $m$ on $(\mathbb N, 2^{\mathbb N})$ by means of a Banach limit and then $m\{n\}=0$ for all $n$ while $m(\mathbb N)=1$.

I am wondering what can be said about merely finitely additive probabilities with countably additive null ideals. What's a typical example of such a probability? "How similar" are such probabilities to countably additive probabilities, i.e. what properties of countably additive probabilities do such probabilities preserve? Any other interesting results about merely finitely additive probabilities with countably additive null ideals are welcome.

Answer 1

Here is an answer for the case that $X$ is countable and all its subsets are measurable.

Let $Y \subset X$ be nonempty, suppose $\{p_y : y \in Y\}$ are strictly positive numbers with $p= \sum_{y \in Y} p_y \le 1.$ Let $\mu$ be an arbitrary finitely additive probability measure on $Y$ (with all subsets measurable) and define a finitely additive probability measure $m$ on $X$ by $$m(A):=(1-p)\mu(A \cap Y)+\sum_{y \in Y \cap A} p_y\, .$$ Then $m$ is a finitely additive probability measure with countably additive null ideal.

Conversely, every finitely additive probability measure $m$ on $X$ with a countably additive null ideal can be obtained this way, by defining $Y:=\{y \in X : m(y)>0\}$ and $p_y=m(y)$ for $y \in Y$ and letting $p= \sum_{y \in Y}$. If $p=1$ then $\mu$ can be arbitrary, while if $p<1$ then take $$\mu(A):=[m(A)- \sum_{y \in A} p_y]/(1-p)\, $$ for $A \subset Y$.

Answer 2

$\newcommand{\N}{\mathbb{N}}\newcommand{\R}{\mathbb{R}}$There are examples on $\R$ with the Borel $\sigma$-algebra $\mathcal{B}$. We take the null ideal to be the meagre Borel sets $\mathcal{M}$ (the $\sigma$-ideal in the Borel sets generated by closed sets with empty interior).

The regular open sets of $\R$ form a complete Boolean algebra $\mathcal{RO}$, and the mapping from $\mathcal{RO} \rightarrow \mathcal{B}/\mathcal{M}$ formed by mapping a regular open set to the equivalence class of Borel sets differing from it by a meagre set is an isomorphism (this uses the Baire category theorem - see for example Fremlin's Measure Theory 514I). What we shall do is define a finitely additive measure $\mu$ on $\mathcal{RO}$ for which the only null element is $\emptyset$. Under the isomorphism above, this defines a finitely-additive Borel probability measure on $\R$ whose null ideal is $\mathcal{M}$.

Let $(U_i)_{i \in \N}$ be a countable base of regular open sets for $\R$ (e.g. open intervals with rational endpoints). By the ultrafilter lemma, for each $i \in \N$, there exists an ultrafilter on $\mathcal{RO}$ containing $U_i$, which defines a finitely-additive measure $\mu_i : \mathcal{RO} \rightarrow [0,1]$ taking only the values $0$ and $1$ and such that $\mu_i(U_i) = 1$.

We then define $\mu : \mathcal{RO} \rightarrow [0,1]$ by $\mu(U) = \sum_{i=1}^\infty 2^{-i} \mu_i(U)$. It is easy to verify that this is a finitely-additive probability measure. Also, for any non-empty regular open $U$ there exists some $i \in \N$ such that $U_i \subseteq U$, and therefore $$ \mu(U) \geq \mu(U_i) \geq 2^{-i}\mu_i(U_i) = 2^{-i} > 0. $$ So the only $\mu$-null regular open set is $\emptyset$.

The measure $\mu$ is not countably additive because on Polish spaces without isolated points there are no countably-additive Borel probability measures that vanish on meagre sets.