This note gathers what is known about, and provides some new results concerning the operations of intersection, of “generated

Let

We are interested in exposing the salient “arithmetical rules” of the operations ∧, ∨, and especially of + and the notion of a complement, delineating their scope through (counter)examples. Apart from pure intellectual curiosity, the justification for the interest in such matters — that may seem a bit “dry” at first — can be seen as coming chiefly from the following observations.

If

[

If

Take again a pair of independent equiprobable signs

[

Concerning the failure of the equality

With the above as motivation, and following the introduction of some further notation and preliminaries in Section

Some general notation and vocabulary. For

Let now

We will indulge in the usual confusion between measurable functions and their equivalence classes mod

A warning: separability per se is not hereditary. For instance

The following basic facts about conditional expectations are often useful; we will use them silently throughout.

For the first claim, by a

We conclude this section with a statement concerning decreasing convergence for martingales indexed by a directed set (it is also true in its increasing convergence guise [

Recall that when

According to [

We begin with some simple observations.

[

[

Let

The next few results deal with the distributivity properties of the pair ∨-∧, when there are strong independence properties.

It is quite agreeable that the preceding statement can be made in such generality. We give some remarks before turning to its proof.

Of course the independence of

The generality of a not necessarily denumerable

Proposition

The inclusion ⊃ in (

[

In

Let

We turn now to complements; we shall resume with the investigation of distributivity later on in Nos.

[

Let

Let

In precise terms, by “discrete”, we mean here, and in what follows, that every

To be precise, by “continuous”, we mean to say here, and in what follows, that every

Let

We have already seen in Example

Even when the equivalent conditions of Proposition

If the equivalent conditions of Proposition

We follow closely the proof of [

Several “stability” properties of conditionally non-atomic

[

We follow closely the proofs of [

The next proposition investigates to what extent complements are “hereditary”.

Dropping, ceteris paribus, the condition that

The situation described by

But there are cases when Proposition

For a less trivial example of the situation described in

Suppose

More generally (in the sufficiency part):

Set

Parallel to Proposition

The converse is not true, because, for instance, one can have

By Proposition

It is clear that

A further substantial statement involving conditional independence and distributivity is the following. It generalizes Proposition

By decreasing martingale convergence,

The generalization to a general

Finally we return yet again to complements. In the following it is investigated what happens if one is given

Let

One would call

Let

Since

We will restrict our attention to the case when there are strong independence properties. A typical example of the type of situation that we have in mind and when the equality

With regard to Remark

Here is now a general result that motivates the investigation of two-sided complements in Proposition

We have

The implication

The author is grateful to an anonymous Referee for providing guidance that helped to improve the presentation of this paper.