Finite additivity, atomlessness and countable additivity

Question

So, I'm trying to get my head around when you can have finitely but not countably additive probabilities.

The standard example of a finitely additive but not countably additive space is the following strange distribution over the natural numbers. All finite sets get measure 0, but the whole space gets measure 1. This is finitely additive but not countably so, since a finite union of finite sets is finite, but a countable union needn't be so.

So this got me thinking that if you had an atomless space, examples of this form would be harder to come by. Does atomlessness plus finite additivity guarantee countable additivity? If not, what is missing?

I know that Villegas (1964) shows that for a comparative probability structure to be countably additive, the important properties of the structure are atomlessness and a certain kind of continuity. But I don't know how relevant that point is to the current question.

Answer 1

Here’s an example.

Define an equivalence relation $\sim$ on $\wp(\Bbb N)$ by $A\sim B$ iff $A\,\triangle\, B$ is finite, where $\triangle$ is symmetric difference, and let $\mathscr{B}=\wp(\Bbb N)/\sim$. For $A\subseteq\Bbb N$ denote by $[A]$ the $\sim$-equivalence class of $A$. Let $\mathscr{U}$ be a free ultrafilter on $\Bbb N$. Note that for any $A\subseteq\Bbb N$, $A\in\mathscr{U}$ iff $[A]\subseteq\mathscr{U}$. Now define a $\{0,1\}$-valued measure $\mu$ on $\mathscr{B}$ by $\mu\big([A]\big)=1$ iff $A\in\mathscr{U}$. Then $\mathscr{B}$ is atomless, and $\mu$ is finitely additive. However, $\mu$ is not countably additive, since it is possible to partition $\Bbb N$ into countably infinitely many infinite sets, none of which is in $\mathscr{U}$.

Added: Michael Greinecker has pointed out that I’m using a notion of atomless that may be different from the one intended by Seamus. Here’s another example that may be preferable.

Let $d:\wp(\Bbb N)\to[0,1]$ be asymptotic density, and let $\mathscr{U}$ be a free ultrafilter on $\Bbb N$. For $A\subseteq\Bbb N$ let $$\mu(A)=\mathscr{U}\text{-}\lim_n\frac{|A\cap\{1,\dots,n\}|}n\;.$$ (For basic information on $\mathscr{U}$-limits see this answer by Martin Sleziak.) Then $\mu$ is a finitely additive non-atomic probability measure on $\wp(\Bbb N)$ such that $\mu(A)=d(A)$ whenever $A$ has an asymptotic density.

Answer 2

The missing ingredient is Monotone Continuity, not atom-less-nes (the villegas 'continuity' condition mentioned by the OP perhaos), as described below, not atom-lessness

Monotone continuity this is the qualitative analog of Kolmogov's continuity condition described above by some of the answers. The CP relation need not be Atom-less, to be countably additive, nor is it a sufficient condition, where atomless, is defined, in the sense of Villegas, (1964, 1967):

$$\forall (A\in S;A>O):\exists (B\in S):(B\subsetneq A)\land A>B>O.$$. where $S$ is the algebra of events, that are under a usually total ordered CP relation $\geq|\leq|$ and $O$ is the impossible event.

To be countable additive, the relation does need to be it needs to be 'monotone continuous' in the sense Villegas (1964): See theorem 2 page 1794 Villegas (1964).

$$(MS)[[A_{n}\uparrow A]\land\forall (n;n\in N^{+}):[B\geq A_{n}]]\iff B\geq A.$$.

Where $B,A\in S$, the algebra of events and $A_n\uparrow A$ is a monotonely increasing sequence of events $A_i\in S$ that convergent onto A

$$A_n\uparrow A\equiv \{A_n\}:[\forall(n):A_{n}\subset A_{n+1}]\,; \cup_{n=1}^{n=\infty}A_{n}=A.$$

This is presumably the 'kind of continuity the OP speaks of.

It is roughly also, entails continuity from above at 0, and thus represent-ability by a countable additive measure in the context of an already numerically representable finitely additive CP relation over a sigma algebra.

Although, I have seen strengthened version in less conventional axiomat-izations which needed to explicitly state the axiom both for continuity from above at $O$ and below and at $\Omega$ where $\Omega$ denotes the certain event.

Otherwise, without monotone continuity, the relation must satisfy something of its ilk, or will not be countably additive full stop.

Over an infinite algebra, the Comparative Probability relation may or may not, be representable by an even finitely additive, depending on what other conditions are satisfied.Dispensing with monotone continuity or something of its kind,unfortunately, is not an option for a CP relation, if it is to be compatible with a countably additive representation.

And even if one desires that the relation to be neither not-atomic, nor defined over a sigma algebra, it requires $(MS)$ but it requires the addition of multiple other axioms $(1)$ and $(2)$in addition to quite possibly losing uniqueness.

-If one dispenses with atomless-ness, then to have a countable additive representation of the CP order, you will need to add $(MS)$,,$(1)$ and $(2)$ to the list of usual axioms below.

• -total comparative CP order. $$A\geq B \vee B\geq C$$
  • strict weak order,(transitivity etc).

- -$$\text{non-negativity and non-triviality}\forall(\Omega_i\in S):\Omega \geq \Omega_i\geq\emptyset$$. -$$\text{non-negativity and non-triviality) }\Omega> \emptyset$$. **-$$\text{monotone continuity}(MS)$$.* -$$(1)\text{perfect separability}$$ -$$(2)\text{Scott's axiom plus archimedean condition}$$*

One will also need , not only $(MS)$, as above, but also $$(1) \text{sigma algebra or perfect separability} \land (2)$$. where,

$(2)$= $(2.2)$ Scott's finite addivity condition with an arch-imedean condition supplement$(2.1)$.

below, or use Scott's infinite condition in addition. Which is much more complex, so I would not take that route.

This is, as non-atomicity allows one to dispense with Scotts, finite condition,(or infinite, or $(2)$ which is the finite Scotts axiom with an archimedean condition. Moreover, one need not have perfect separability ,I think, which non-atomicy entails, in the case of a non-sigma algebra, countably addive but non , non-atomic representation. I am not going the type this (Scott's infinite condition )axiom here, it is rather complicated (see Z. Domotor's 1969 PHD thesis (Probalistic relational structures; see https://www.google.com.au/search?q=domotor+relational+systems&ie=utf-8&oe=utf-8&client=firefox-b&gfe_rd=cr&dcr=0&ei=dAfnWbDAOeLc8weYo5XQBQ

or use the conditions mentioned by Jaffray and Chauttenaff (1984) which are those of $(1)$ and $(2)$ below and discussed in "Archimedean Qualitative Probabilities", listed below in the references.

Roughly,for a countable additive (non, non-atomic representation) that $(1)$ and $(2)$ must be added to the usual axioms. As well as monotone continuity $(MS)$.

$$(1)$$

$(1)$ keeping the sigma algebra condition of Villegas or: $(1a)$replace the the $\sigma$ algebra,with a 'perfect separability ' axiom, and possibly a strengthened archi-medean condition. $(2)$ replacing De-finetti additivity: $(F)$.

$(2)$ $$(F):A\cap C=B\cap C=\emptyset: A\geq B \iff A \cup C \geq B \cup C$$.

with a supplemented version of Dana Scott's axiom/strong additivity, finite cancellation (strong additivity) . That is supplemented with an addition archimedean constraint.

.

where condition (2) mentioned below is called (A) in the picture from Qualitative Archimdedean PRobabilites (1984).

(2) or (A) in the cited picture, is the $(2.1)$ Archimedean axiom and $(2.1)$ Scotts axiom when combined into one condition See (chautenaff 1984) referenced below for more on this.

It is, the supplemented Scott's(1964) condition $(A)$ in Chauttenauff (1984).

I have posted a link and picture below. $$(3)$\text{add monotone continuity }MS$$.

Or you use Scott's infinite axiom (I am not sure if you want to do that)!!!

Moreover, if you dispense with both the sigma algebra requirement, and atomless-ness, then you will need in addition to the above, a perfect separability requirement atomless-ness is not sufficient for a countably additive representation. It almost always is necessary for a unique representation.

-The sigma algebra relation can be dispensed with as well, but unless one wishes to add perfect separability.

If you want to know the conditions that are necessary and sufficient for a non-necessarily unique countably additive representation, read Jaffray and chautenaff (1984) who give simpler set of axioms for a comparative probability structure that is necessary and sufficient for a countably additive representation, in particular a representation over an infinite sample space.

If one dispenses with atom-less-ness (in the traditional comparative probability sense; that is in the sense of Villegas (1964), then $(1)$ one does require additional axioms to those of Villegas,

$(2)$ In addition, the representation will likely no longer be unique.

For example, if one dispenses, with both the sigma algebra condition and atom-less-ness:

-(1)(then one must replace) s de-finetti's axiom in Villegas set of axioms: $$A\cap C=B\cap C=\emptyset: A\geq B \iff A U C \geq B UC$$. with $(A),\land (B),

-If one keeps the sigma algebra condition

• Then perfect separability is entailed, by one needs only (1), and (3)

• See, Jaffray 'On the Extension of Additive Utilities to Infinite Sets'(1974) which are necessary and sufficient space satisfy. Is another on not necessarily non-atomic or countably additive probability measures that may not even be on sigma algebras.

One good source is Fisburne (1986), the axioms of subjective probability

Fishburn, Peter C., [The axioms of subjective probability]; see (http://dx.doi.org/10.1214/ss/1177013611), Stat. Sci. 1, 335-358 (1986). ZBL0604.60004. Theories of probability), Luce, Narens; maybe even fisburne 1986).

Villegas, C., see

1. On qualitative probability $\sigma$-algebras, Ann. Math. Stat. 35, 1787-1796 (1964). ZBL0127.34807. but it must be monotone continuous (or something like it).
2. Villegas, C., On qualitative probability, Am. Math. Mon. 74, 661-669 (1967). ZBL0154.42301. 3.Chateauneuf, Alain; Jaffray, Jean-Yves, [Archimedean qualitative probabilities].

(http://dx.doi.org/10.1016/0022-2496(84)90026-9), J. Math. Psychol. 28, 191-204 (1984). ZBL0558.60003. if you retain the sigma algebra requirement, but forsake Atom-less-ness then in addition to possibility losing 'unique-ness':