A necessary and sufficient condition on a sequence

Ever since the appearance of the Martingales Convergence Theorem, there have been attempts to find criteria for a larger group of sequences of

In this paper we will give necessary and sufficient conditions for almost-everywhere convergence of

Conditional expectations convergence theorems have many applications in probability theory, data science, economics, finance, statistics and give alternative proofs of theorems such as Kolmogorov large-numbers, Levy generalized Borel–Cantelli, Radon–Nikodym. For potential applications, see [

Let

In [

The quest of this paper was to see if we could establish somewhat similar conditions on the

To deal with the other part of the problem, we first characterize the conditions through a set

It has been pointed out by one of the reviewers that in the literature the results are usually referred to

As usual, we will use the notation

In [

Another approach was given by Fetter [

Since the condition of the above theorem is fulfilled for monotone sequences of

Given a sequence

Now, since these

Finally we point out that none of the theorems but Theorem

In [

It is clear that in order to have convergence a.e. we need a stronger concept for sets in

We begin by establishing the following lemma.

Notice that

To prove

Using the same argument for

Since

We will prove now that

Then, by hypothesis,

Finally, to prove that

Let

The relationship between the measure of a set and the above norm will be shown in the appendix. We prove there that for

We say that

The next lemma shows that actually we can relax the condition

By Lemma

Thus, for

In general, for

Let

Now, since

We will say that a sequence of sets

It is easily seen that, if

Let

Since

A similar argument can be used for the case of the union of two sets. □

Let

In view of the proof, we can actually relax somewhat the condition of uniformly covering to get a similar result.

The proof is exactly the same as that of the above lemma. □

Notice that if A is uniformly covered by

The interesting case occurs when both

Since

Now we state the necessary lemma for a.e. convergence of characteristic functions.

Let

Therefore,

It is also clear that

Thus,

Finally, since

Combining Lemma

In view of the above theorem it is clear that it is convenient to establish the following definition.

Given a sequence of

Notice the following lemma.

Taking

Notice that if

Let

So

To prove

For

We also have

And so by Lemma

How do the above results look in the case of

In [

Notice that, since we defined

Now we give the following definition.

We have the following result.

Let

Thus, if

That is, there is an

Let

Let

We have now constructed a sequence

In [

The following proposition is an immediate property.

Notice that in this case

Two properties of this

Before we proceed we will study a special case.

We call the sequence

The main property of these 2-bounded sequences of

In view of Lemma

Of course, the 2-boundeness condition is a very strong restriction. However, we would like to point out that any increasing or decreasing sequence of

We will see that the cases for which

Two immediate consequences follows.

Notice that by Theorem

To prove

We start the proof of the case

We are going to prove now that

But by (

This means that

The property

Notice that in the case of a.e. convergence,

We will show the relation between the measure of a set and the

Let

Therefore, for any

To prove the equality, notice that we can assume

By definition the set

We would like to thank Anthony Torres-Hernandez, Universidad Pomeu-Fabre and William Gutierrez, Universidad San Carlos Guatemala for their always useful comments. We are also grateful to the referees for their observations and suggestions.

This work was partially funded by the National University of Mexico project UNAM-PAPIIT 33-IT101421.