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We study the structure of the ergodic limit functions determined in random ergodic theorems. When the

The present paper is concerned with the relations between the limit functions in random ergodic theorems and the random parameters concomitant to the limit functions. The first results of the random ergodic theory include Pitt’s random ergodic theorem [

It was pointed out by Marczewski (see [

In what follows, we suppose that there are a given

It seems to be worthwhile to include the first random ergodic theorems which may be stated, respectively, as follows.

Let

Exactly speaking, the first step to the theory of random ergodic theorems was taken by Pitt [

Let

Kakutani’s paper on the random ergodic theorem was published in 1950, but the random ergodic theorem had already been dealt with by Kawada (“Random ergodic theorems”, Suritokeikenkyu (Japanese) 2, 1948). Later, Kawada reminisced about the circumstances of an affair of his paper. It is the rights of matter that Kawada’s result is due to Kakutani’s kind suggestion.

In Kakutani’s random ergodic theorem (as well as in Pitt-Ulam-von Neumann’ theorem), the sequence

The most general formulation of random ergodic theorems is the following Chacon’s type theorem given by Jacobs [

If all

Throughout all that follows, let

Let

Then,

Our main result is stated as follows.

Let

We need the following lemmas.

Lemma 5 (see [

Using the measurable version appearing in Lemma 5, we define

From the norm conditions of

Moreover, as is easily checked, we find that

One can easily verify that excepting a suitable

It is clear that if

Lemma 6. It holds

Thus, excepting a

We return to the proof of the theorem. By Lemma 6 and the martingale convergence theorem (cf. [

If we take

Let

As before, we define

Now, adapting the (almost) same argument as used in the proof of Theorem

In the setting of measure-preserving transformations, Theorem

Let

As already seen above, we can define the operators

In particular, if the operators in question are commutative then we have the following.

Let

In passing, we make mention of the

Let

A sub-Markovian operator

Now, let

In addition, if

Let the measure

Since

Now, the above general theorems can also be applied to sub-Markovian operators. For example, Theorem

Let

In view of Lemma

In this section, we establish a

Let

Note here that in Theorem

Let

Define the skew product

Let

For example, if

In particular, applying Irmisch’s theorem to sub-Markovian operators, we have the following.

Let

In view of Lemma

The relations between the random ergodic limit functions and the random parameters have been investigated (with satisfactory formulations) only in discrete parameter cases so far. So, it is very interesting to study the continuous analogs of the theorems obtained above. But no continuous results are known from the point of view of the dependence of the limit functions on the random parameters. Here, it is worthwhile to notice that Anzai has obtained a continuous version of Kakutani’s random ergodic theorem for Brownian motion in continuous parameter cases (see [

Let

The random ergodic theorems obtained above can be applied to the nonlinear random ergodic theorems for affine systems (see [

Let

It follows that there exists a

As far as we are concerned with the ergodic behaviors of Cesàro-type processes for nonexpansive operators on

Let

Let

The following example given by Gładysz [

In this example, we consider the measure spaces

In the setting of Example

It is an interesting problem to ask what happens if we transform a function

This is an immediate consequence of the ergodicity of the family

The author is certainly indebted and very grateful to Mrs. Caroline Nashat for her kind assistance.