2.1 Integration on finitely additive spaces

We review the theory of integration on finitely additive probability spaces and recall the properties we need in this paper. Our exposition closely follows the seminal monograph by K. Bhaskara Rao and M. Bhaskara Rao [Reference Bhaskara Rao and Bhaskara Rao5].

The first step is to define the integral of simple functions.

Definition 2.1. A real-valued function f on $\Omega $ is said to be simple if there exist sets $\{A_i\}_{i=1}^n \subseteq \mathcal {F}$ forming a partition of $\Omega $ and real numbers $a_i \in \mathbb {R}$ such that for all $\omega \in \Omega $ ,

$$ \begin{align*} f(\omega) = \sum_{i=1}^n a_i I_{A_i}(\omega). \end{align*} $$

For such a function f, the integral with respect to $\mu $ is defined by

$$ \begin{align*} \mu(f) := \sum_{i=1}^n a_i \mu(A_i). \end{align*} $$

It is straightforward to verify that $\mu (f)$ is independent of the particular representation of f. Moreover, linear combinations of simple functions are simple, and the absolute value of a simple function is simple (see [Reference Bhaskara Rao and Bhaskara Rao5, Propositions 4.4.2 and 4.4.4]).

As in the $\sigma $ -additive case, a general integrable function $f : \Omega \to \mathbb {R}$ will be defined as the limit of simple functions. In the $\sigma $ -additive case, taking this limit to be almost surely or in probability leads to equivalent definitions. In the finitely additive case, we adopt convergence in probability.

Definition 2.2. (Cf. [Reference Bhaskara Rao and Bhaskara Rao5, Definition 4.3.1])

A sequence $\{f_n\}$ of real-valued functions on $\Omega $ is said to converge hazily to a function f if

$$ \begin{align*} \Big(\forall \varepsilon> 0, \ \Big(\lim_{n \to \infty} \mu^*(\{\omega \in \Omega : |f_n(\omega) - f(\omega)| > \varepsilon \}) = 0\Big)\Big), \end{align*} $$

where $\mu ^*:\mathcal {P}(\Omega ) \to [0,1]$ denotes the outer measure of $\mu $ , defined by

$$ \begin{align*} \mu^*(A) := \inf\{\mu(B) : A \subseteq B,\, B \in \mathcal{F}\}. \end{align*} $$

We can now define the integral for general functions.

Definition 2.3. (Cf. [Reference Bhaskara Rao and Bhaskara Rao5, Definition 4.4.11])

A real-valued function f on $\Omega $ is said to be integrable if there exists a sequence $\{f_n\}$ of simple functions such that $\{f_n\}$ converges hazily to f and

$$ \begin{align*} \lim_{n,m \to \infty} \mu(|f_n - f_m|) = 0. \end{align*} $$

Such a sequence $\{f_n\}$ is called a determining sequence for f. The integral of f is then defined by

$$ \begin{align*} \mu(f) := \lim_{n \to \infty} \mu(f_n). \end{align*} $$

Furthermore, we define

$$ \begin{align*} L_1(\Omega, \mathcal{F}, \mu) := \{ f : \Omega \to \mathbb{R} \mid |f| \text{ is integrable} \} \end{align*} $$

and write $\lVert {f} \rVert _1$ for $\mu (\vert f\vert )$ .

The fact that the above definition of the integral is well defined (that is, independent of the choice of determining sequence and finite) follows from [Reference Bhaskara Rao and Bhaskara Rao5, Proposition 4.4.10].

Furthermore, the finitely additive integral satisfies the expected fundamental structural properties (cf. [Reference Bhaskara Rao and Bhaskara Rao5, Proposition 4.4.13]). In particular, linear combinations of integrable functions are themselves integrable, the integral defines a linear operator on the space of integrable functions, and it is monotone: if $f,g$ are integrable and $f(\omega ) \le g(\omega )$ for all $\omega \in \Omega $ , then $\mu (f) \le \mu (g)$ .

The final result we require concerning integration on finitely additive probability spaces involves the composition of integrable functions with measure-preserving automorphisms. Specifically, we consider an invertible map $\tau \colon \Omega \to \Omega $ satisfying $\tau ^{-1}(A) \in \mathcal {F}$ and $\mu (\tau ^{-1}(A)) = \mu (A)$ for all $A \in \mathcal {F}$ .

Proposition 2.1. If $f : \Omega \to \mathbb {R}$ is integrable and $\tau : \Omega \to \Omega $ a measure-preserving automorphism, then the function $\tilde {f} : \Omega \to \mathbb {R}$ defined by

$$ \begin{align*} \tilde{f}(\omega) := f(\tau(\omega)) \end{align*} $$

is integrable and $\mu (\tilde{f}) = \mu (f)$ .

Proof. The result is immediate if f is simple. Let $\{{f_n}\}$ be a determining sequence for f and define $\{{\tilde{f_n}}\}$ by

$$ \begin{align*} \tilde{f_n}(\omega) := f_n(\tau(\omega)). \end{align*} $$

Since $\{{f_n}\}$ and $\{{\tilde{f_n}}\}$ are sequences of simple functions, and $\tau $ is measure preserving, we have

$$ \begin{align*} \mu(f_n) = \mu(\tilde{f_n}) \quad \text{for all } n, \end{align*} $$

and, hence, if $\{{\tilde{f_n}}\}$ is a determining sequence for $\tilde{f}$ , we have

$$ \begin{align*} \mu(\tilde{f}) = \lim_{n \to \infty} \mu(\tilde{f_n}) = \lim_{n \to \infty} \mu(f_n) = \mu(f). \end{align*} $$

Thus, it remains to show that $\{{\tilde{f_n}}\}$ is indeed a determining sequence for $\tilde{f}$ .

We first note the following.

Claim. If $\tau $ is a measure-preserving automorphism, then for every $E \subseteq \Omega $ ,

$$ \begin{align*} \mu^*(\tau^{-1}(E)) = \mu^*(E). \end{align*} $$

Proof of claim.

For any $B \in \mathcal {F}$ with $E \subseteq B$ , we have $\tau ^{-1}(E) \subseteq \tau ^{-1}(B)$ and $\mu (\tau ^{-1}(B)) = \mu (B)$ . Taking the infimum over all such B yields

$$ \begin{align*} \mu^*(\tau^{-1}(E)) \le \inf\{\mu(\tau^{-1}(B)) : E \subseteq B,\, B \in \mathcal{F}\} = \mu^*(E). \end{align*} $$

The reverse inequality follows by applying the same argument to $\tau ^{-1}$ .

Now, for all $\varepsilon> 0$ and $n \in \mathbb {N}$ ,

$$ \begin{align*} \begin{aligned} \mu^*(\{\omega : |\tilde{f_n}(\omega) - \tilde{f}(\omega)|> \varepsilon \}) &= \mu^*(\tau^{-1}(\{\omega : |f_n(\omega) - f(\omega)| > \varepsilon \}))\\ &= \mu^*(\{\omega : |f_n(\omega) - f(\omega)| > \varepsilon \}). \end{aligned} \end{align*} $$

Thus, since $\{{f_n}\}$ converges hazily to f, it follows that $\{{\tilde{f_n}}\}$ converges hazily to $\tilde{f}$ .

Finally, for all $n,m \in \mathbb {N}$ ,

$$ \begin{align*} \mu(|\tilde{f_n} - \tilde{f_m}|) = \mu(|f_n - f_m|), \end{align*} $$

and, hence,

$$ \begin{align*} \lim_{n,m \to \infty} \mu(|\tilde{f_n} - \tilde{f_m}|) = \lim_{n,m \to \infty} \mu(|f_n - f_m|) = 0. \end{align*} $$

This shows that $\{{\tilde{f_n}}\}$ is indeed a determining sequence for $\tilde{f}$ , completing the proof.

2.2 Quantitative notions of stochastic convergence

Quantitative aspects of the pointwise ergodic theorem have been studied extensively in the $\sigma $ -additive setting. As we shall see in the following section, many of these quantitative results carry over to the finitely additive setting since one can establish a version of the Calderón transference principle [Reference Calderón7] for finitely additive spaces. In this section, we review the relevant quantitative notions of almost sure convergence that are pertinent to our work.

The most natural quantitative interpretation of the convergence of a sequence of real numbers $\{{x_n}\}$ is known as a rate of convergence, which is a function $\phi :(0,1) \to \mathbb {R}^+$ satisfying

$$ \begin{align*} (\forall \varepsilon \in (0,1), \forall i,j \ge \phi(\varepsilon),\ (|x_i - x_j| \le \varepsilon)). \end{align*} $$

There are many convergence results in analysis where one cannot obtain a uniform rate of convergence for the sequences satisfying the premises of these results; a notable example is the monotone convergence theorem, where no uniform rate of convergence exists for all monotone sequences in $[0,1]$ , since such sequences can converge arbitrarily slowly. This remains true even for the class of computable sequences of rational numbers [Reference Specker36].

In situations where a uniform rate of convergence cannot be obtained, one can often obtain uniform bounds on an equivalent reformulation, known as metastable convergence:

(2.1) $$ \begin{align} (\forall \varepsilon>0, \forall g:\mathbb{N}\to \mathbb{N}, \ \exists n \ \forall i,j\in [n;n+g(n)]\, (|x_i-x_j|\leq \varepsilon)). \end{align} $$

It is easy to see that (2.1) is equivalent to the convergence of $\{{x_n}\}$ . Indeed, if $\{{x_n}\}$ does not converge, then there exists some $\varepsilon>0$ such that

$$ \begin{align*} (\forall n,\ \exists i,j\ge n \, (|x_i-x_j|> \varepsilon)), \end{align*} $$

so we can take a function $g:\mathbb {N}\to \mathbb {N}$ bounding $i,j$ depending on n, that is,

$$ \begin{align*} (\forall n,\ \exists i,j \in [n;n+g(n)]\, (|x_i-x|> \varepsilon)), \end{align*} $$

which contradicts (2.1). The converse implication is immediate.

A rate of metastability is a functional $\Phi $ bounding n in (2.1), that is,

$$ \begin{align*} & (\forall \varepsilon \in (0,1), \forall g:\mathbb{N}\to \mathbb{N},\ \exists n \le \Phi(\varepsilon,g), \forall i,j\in [n;n+g(n)], \ (|x_i-x_j|\leq \varepsilon)). \end{align*} $$

Remark 2.2. One can easily verify that if $\{{x_n}\} \subseteq [0,1]$ is monotone, then

$$ \begin{align*} \Phi(\varepsilon,g) := \tilde g^{(\lceil 1/\varepsilon \rceil)}(0) \end{align*} $$

is a rate of metastability for $\{{x_n}\}$ , where $\tilde g(n) := n + g(n)$ and $\tilde g^{(i)}$ denotes the ith iteration of $\tilde g$ .

In the stochastic setting, the analog of a rate of convergence is a function $\phi : (0,1) \times (0,1) \to \mathbb {R}^+$ satisfying

$$ \begin{align*} & (\forall \varepsilon,\ \unicode{x3bb} \in (0,1)\\ & \quad \forall k \in \mathbb{N},\ (\mu(\forall i,j \in [\phi(\unicode{x3bb},\varepsilon);\phi(\unicode{x3bb},\varepsilon)+k] (|X_i - X_j| \le \varepsilon))> 1 - \unicode{x3bb})), \end{align*} $$

where $\{{X_n}\}$ is a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon \in (0,1),\ (\{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})). \end{align*} $$

Such a function is sometimes called a rate of almost sure convergence. In [Reference Avigad, Gerhardy and Towsner3, Theorem 5.1], it is demonstrated that no computable such rate can exist for the convergence of the ( $\sigma $ -additive) pointwise ergodic theorem. Motivated by the deterministic case, the notion of uniform metastable convergence (there is also a notion of pointwise metastable convergence, cf. Remark 3.5) was introduced in [Reference Avigad, Gerhardy and Towsner3].

Definition 2.4. Let $\{{X_n}\}$ be a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon>0,\ (\{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})). \end{align*} $$

We say that $\{{X_n}\}$ is uniformly metastable if

$$ \begin{align*} & (\forall \varepsilon,\unicode{x3bb}>0, \forall g:\mathbb{N}\to \mathbb{N},\ \exists n\in \mathbb{N} \\ & \quad( \mu(\forall i,j \in [n,n+g(n)] \, (|X_i - X_j| \le \varepsilon)) > 1 - \unicode{x3bb})). \end{align*} $$

As in the deterministic case, uniform metastability is equivalent to finite almost sure convergence.

Proof. Suppose first that $\{X_n\}$ is finitely almost surely convergent. Then, for every $\varepsilon , \unicode{x3bb}> 0$ , there exists $N \in \mathbb {N}$ such that

$$ \begin{align*} (\forall k \in \mathbb{N}, \, (\mu(\forall i,j \in [N;N+k]\, (|X_i - X_j| \le \varepsilon))> 1 - \unicode{x3bb})). \end{align*} $$

Let $g \colon \mathbb {N} \to \mathbb {N}$ be arbitrary. Taking $k = g(N)$ in the above yields

$$ \begin{align*} \mu(\forall i,j \in [N;N+g(N)]\, |X_i - X_j| \le \varepsilon)> 1 - \unicode{x3bb}, \end{align*} $$

which shows that $\{{X_n}\}$ is uniformly metastable.

Conversely, suppose $\{{X_n}\}$ is not finitely almost surely convergent. Then, there exist $\varepsilon , \unicode{x3bb}> 0$ such that for all $N \in \mathbb {N}$ , there exists $k \in \mathbb {N}$ with

$$ \begin{align*} \mu(\forall i,j \in [N;N+k]\, |X_i - X_j| \le \varepsilon) \le 1 - \unicode{x3bb}. \end{align*} $$

Define $g(N)$ to be the least such k. Then, for this choice of g, we have, for all $N\in \mathbb {N}$ ,

$$ \begin{align*} \mu(\forall i,j \in [N;N+g(N)]\, |X_i - X_j| \le \varepsilon) \le 1 - \unicode{x3bb}, \end{align*} $$

which shows that $\{{X_n}\}$ fails to be uniformly metastable.

To prove Theorem 1.3, we shall demonstrate that the ergodic averages are uniformly metastable. Furthermore, we shall obtain a quantitative strengthening of this result through the construction of explicit so-called rates of uniform metastability.

Definition 2.5. Let $\{{X_n}\}$ be a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \text{ for all } \varepsilon>0,\ (\{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})). \end{align*} $$

A functional $\Phi : (0,1)\times (0,1)\times \mathbb {N}^{\mathbb {N}} \to \mathbb {R}^+$ is called a rate of uniform metastability for $\{{X_n}\}$ if

$$ \begin{align*} &(\forall \varepsilon,\unicode{x3bb} \in (0,1), \forall g:\mathbb{N}\to \mathbb{N}, \ \exists n \le \Phi(\unicode{x3bb},\varepsilon,g)\, (\mu(\forall i,j \in [n,n+g(n)] \, (|X_i - X_j| \le \varepsilon))> 1 - \unicode{x3bb})). \end{align*} $$

In [Reference Avigad, Gerhardy and Towsner3], rates of uniform metastability were computed for the pointwise ergodic theorem for $L_2$ functions, with the treatment of the full theorem later given by the author and Powell in [Reference Neri and Powell29]. Moreover, in [Reference Neri and Powell29], a stronger quantitative result is established by introducing the concept of uniform learnability.

Definition 2.6. Let $\{{X_n}\}$ be a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon>0,\ (\{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})). \end{align*} $$

A function $\phi :(0,1)\times (0,1)\to \mathbb {R}^+$ is called a learnable rate of uniform convergence if, for any $\varepsilon , \unicode{x3bb} \in (0,1)$ , and sequences $a_0<b_0 \le a_1<b_1 \le \cdots $ , there exists $n \le \phi (\unicode{x3bb} ,\varepsilon )$ such that

$$ \begin{align*} \mu(\exists i,j\in [a_n,b_n] \ |X_i-X_j|\geq \varepsilon ) \le \unicode{x3bb}. \end{align*} $$

It turns out that a learnable rate of uniform convergence corresponds to a rate of metastability of a particularly nice form.

Proposition 2.6. (Cf. [Reference Neri and Powell29])

Let $\{{X_n}\}$ be a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon>0\ (\{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})). \end{align*} $$

Then,

$$ \begin{align*} \Phi(\unicode{x3bb},\varepsilon,g) = \tilde g^{(\lceil \phi(\unicode{x3bb},\varepsilon) \rceil)}(0) \quad \text{with } \tilde g(n) := n + g(n) \end{align*} $$

is a rate of uniform metastability for $\{{X_n}\}$ if and only if $\lceil \phi (\unicode{x3bb} ,\varepsilon ) \rceil $ is a learnable rate of uniform convergence.

This result is stated in the context of $\sigma $ -additive spaces, immediately following [Reference Neri and Powell29, Definition 4.15], as an easy consequence of an abstract result given in [Reference Neri and Powell29, Lemma 3.5]. The finitely additive case follows similarly as a direct consequence of the same lemma.

We prove a quantitatively stronger result than that stated in Theorem 1.3 by computing a learnable rate of uniform convergence for the ergodic averages. Theorem 1.3 will then follow from Propositions 2.5 and 2.6.

To compute a learnable rate of uniform convergence for the pointwise ergodic theorem, we require a result from [Reference Jones, Kaufman, Rosenblatt and Wierdl12] concerning the fluctuations of ergodic averages on $\sigma $ -additive probability spaces. First, we introduce the notion of the number of $\varepsilon $ -fluctuations of a sequence of real numbers.

Definition 2.7. Let $\{{x_n}\}$ be a sequence of real numbers. We define $J_{{N},{\varepsilon }}\{{x_n}\}$ to be the number of $\varepsilon $ -fluctuations that occur in the initial segment $\{x_0,\ldots ,x_{N-1}\}$ , that is, the maximal $k \in \mathbb {N}$ such that there exist indices

$$ \begin{align*} i_1 < j_1 \le i_2 < j_2 \le \cdots \le i_k < j_k < N \end{align*} $$

with

$$ \begin{align*} |x_{i_l}-x_{j_l}| \ge \varepsilon \quad \text{for all } l=1,\ldots,k. \end{align*} $$

We write

$$ \begin{align*} J_{{\varepsilon}}\{{x_n}\} = \lim_{N\to\infty} J_{{N},{\varepsilon}}\{{x_n}\} \end{align*} $$

for the total number of $\varepsilon $ -fluctuations that occur in $\{{x_n}\}$ . Note that $J_{{\varepsilon }}\{{x_n}\}$ could be infinite.

Remark 2.7. It is clear that $\{{x_n}\}$ converges if and only if

$$ \begin{align*} (\forall \varepsilon> 0, (J_{{\varepsilon}}\{{x_n}\} < \infty)). \end{align*} $$

In [Reference Jones, Kaufman, Rosenblatt and Wierdl12], Jones et al obtained the following quantitative strengthening of the pointwise ergodic theorem.

Theorem 2.8. Let $(\Omega , \mathcal {F},\mu )$ be a $\sigma $ -additive probability space and let $\tau :\Omega \to \Omega $ be a measure-preserving automorphism. For any $f \in L_1$ and $k, \varepsilon> 0$ , we have

$$ \begin{align*} \mu(J_{{\varepsilon}}\{{A_nf}\} \ge k) \le \frac{C \lVert {f} \rVert_1}{\varepsilon \sqrt{k}} \end{align*} $$

for some numerical constant C.

This result was originally conjectured by Ivanov [Reference Kachurovskii13], who, using entirely different methods, was only able to show

$$ \begin{align*} \mu(J_{{\varepsilon}}\{{A_nf}\} \ge k) \le C \sqrt{\frac{\log(k)}{k}}, \end{align*} $$

where C is a constant depending only on $\lVert {f} \rVert _1/\varepsilon $ (in [Reference Neri and Powell29], the authors were unaware of the bound of Jones, Kaufman, Rosenblatt, and Wierdl, and by improving the methods of Ivanov, they were able to give an improvement to his bound, which was still weaker than that in Reference Jones, Kaufman, Rosenblatt and Wierdl12]).

By establishing a version of the Calderón transference principle [Reference Calderón7] for finitely additive spaces, we shall show that a version of Theorem 2.8 also holds in the finitely additive setting.

It is clear from Theorem 2.8 that the ergodic averages satisfy

$$ \begin{align*} (\forall \varepsilon> 0, \ (J_{{\varepsilon}}\{{A_nf}\} < \infty \text{ almost surely})), \end{align*} $$

and thus converge almost surely. However, as in the case of convergence, bounded fluctuations can have different, non-equivalent formulations in the finitely additive setting. If $\mathcal {F}$ is a $\sigma $ -algebra and $\{{X_n}\}$ is a sequence of random variables on $\Omega $ (so that for all k and $\varepsilon $ , $J_{{k},{\varepsilon }}\{{X_n}\}$ is itself a random variable), we have the following formulations of having finitely many fluctuations almost surely:

1. (J1) $\mu (\{\omega \in \Omega : \forall \varepsilon> 0,\, \exists N \in \mathbb {N}, \forall k\in \mathbb {N}\, J_{{k},{\varepsilon }}\{{X_n}\}(\omega )< $ $ N\}) = 1;$

2. (J2) $(\forall \varepsilon> 0,\ (\mu (\{\omega \in \Omega : \exists N\in \mathbb {N} \,\forall k \in \mathbb {N}\, (J_{{k},{\varepsilon }}\{{X_n}\}(\omega ) < N)\}) = 1));$

3. (J3) $(\forall \varepsilon , \unicode{x3bb}> 0,\ \exists N\in \mathbb {N}\, (\mu (\{\omega \in \Omega : \forall k\in \mathbb {N}\, (J_{{k},{\varepsilon }}\{{X_n}\}(\omega ) < N)\})> 1 - \unicode{x3bb} );$

4. (J4) $(\forall \varepsilon , \unicode{x3bb}> 0,\ \exists N\in \mathbb {N}\ \forall k\in \mathbb {N}\, (\mu (\{\omega \in \Omega : J_{{k},{\varepsilon }}\{{X_n}\}(\omega ) < N\})> 1 - \unicode{x3bb} ).$

As in the case of convergence, we have

$$ \begin{align*} (\mathrm{J}1) \Rightarrow (\mathrm{J}2), \quad (\mathrm{J}3) \Rightarrow (\mathrm{J}2), \quad \text{and} \quad (\mathrm{J}3) \Rightarrow (\mathrm{J}4). \end{align*} $$

Furthermore, it is clear that

$$ \begin{align*} (\mathrm{J2}) \iff (\mathrm{B}), \end{align*} $$

where formulation (B) is the almost sure convergence given by

$$ \begin{align*} &(\forall \varepsilon> 0, ( \mu(\{\omega \in \Omega : \exists N \in \mathbb{N}\, \forall i,j \ge N\, (|X_i(\omega) - X_j(\omega)| \le \varepsilon) \}) = 1)). \end{align*} $$

Therefore, Example 1.1 demonstrates that the pointwise ergodic theorem cannot satisfy formulation (J2) in the finitely additive setting and thus cannot satisfy formulation (J3). However, Theorem 2.8 implies formulation (J3), so it cannot hold in the finitely additive setting. We shall show that formulation (J4) of Theorem 2.8 does hold via the Calderón transference principle [Reference Calderón7] for finitely additive spaces.

Theorem 1.3 will follow from our formulation (J4) of Theorem 2.8 by establishing a quantitative connection between bounds on the fluctuations and learnable rates of uniform convergence. We first introduce the notion of a modulus of finite fluctuations, as in [Reference Neri and Powell29], which quantitatively captures formulation (J4).

Definition 2.8. Let $\{{X_n}\}$ be a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon>0,\ (\{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})). \end{align*} $$

A function $\phi :(0,1)\times (0,1)\to \mathbb {R}^+$ is a modulus of finite fluctuations for $\{{X_n}\}$ if

$$ \begin{align*} (\forall \varepsilon, \unicode{x3bb} \in (0,1), \forall k\in \mathbb{N}, \ (\mu(J_{{k},{\varepsilon}}\{{X_n}\} < \phi(\varepsilon,\unicode{x3bb}))> 1 - \unicode{x3bb})). \end{align*} $$

Remark 2.9. This definition is well posed because, if

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon>0,\ ( \{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F})), \end{align*} $$

then

$$ \begin{align*} (\forall k \in \mathbb{N}, \forall \varepsilon, N> 0,\ (\{\omega \in \Omega : J_{{k},{\varepsilon}}\{{X_n}\}(\omega) < N\} \in \mathcal{F})). \end{align*} $$

In [Reference Neri and Powell29], the author and Powell examined how quantitative notions of almost sure convergence, including those introduced here, relate to one another. For example, from a learnable rate of uniform convergence, one can obtain a rate of uniform metastability. Furthermore, the authors abstracted away from almost sure convergence and instead focused on general logical formulas of the same quantifier complexity, yielding more general quantitative results.

At the time of writing [Reference Neri and Powell29], the quantitative relationship between moduli of finite fluctuations and learnable rates of uniform convergence in the $\sigma $ -additive setting was left as an open problem. Recently, Powell communicated to the author a partial solution to this problem by constructing a learnable rate of uniform convergence from a modulus of finite fluctuations. Powell’s construction also lifts naturally to the finitely additive setting. Namely, one has the following result.

Proposition 2.10. Let $\{{X_n}\}$ be a sequence of real-valued functions on $\Omega $ satisfying

$$ \begin{align*} (\forall i,j \in \mathbb{N}, \forall \varepsilon>0,\ \{\omega \in \Omega: |X_i(\omega)-X_j(\omega)|\le \varepsilon\} \in \mathcal{F}). \end{align*} $$

Suppose $\{{X_n}\}$ has a modulus of finite fluctuations $\phi $ . Then, $\{{X_n}\}$ has a learnable rate of uniform convergence given by

$$ \begin{align*} \psi(\unicode{x3bb},\varepsilon) := \frac{2 \, \phi(\unicode{x3bb}/2,\varepsilon)}{\unicode{x3bb}}. \end{align*} $$

Proof. Fix $\unicode{x3bb} , \varepsilon \in (0,1]$ and consider sequences $a_0 < b_0 \le a_1 < b_1 \le \cdots $ . For $k \in \mathbb {N}$ , define the event

$$ \begin{align*} B_k := \{\omega \in \Omega : J_{{k},{\varepsilon}}\{{X_n}\}(\omega) \ge \phi(\unicode{x3bb}/2,\varepsilon)\} \in \mathcal{F}. \end{align*} $$

By the definition of the modulus of finite fluctuations, $\mu (B_k) \le \unicode{x3bb} /2$ for all $k \in \mathbb {N}$ . For $a,b \in \mathbb {N}$ , set

$$ \begin{align*} C(a,b) := \{\omega \in \Omega : \exists i,j \in [a,b], \ |X_i(\omega)-X_j(\omega)|> \varepsilon\} \in \mathcal{F}. \end{align*} $$

Suppose, for a contradiction, that for all $n \le \psi (\unicode{x3bb} ,\varepsilon )$ , we have $\mu (C(a_n,b_n))> \unicode{x3bb} $ . Then, for all $n \le \psi (\unicode{x3bb} ,\varepsilon )$ ,

$$ \begin{align*} \mu(C(a_n,b_n) \cap B_k^c) = \mu(C(a_n,b_n)) - \mu(C(a_n,b_n) \cap B_k)> \unicode{x3bb} - \mu(B_k) \ge \frac{\unicode{x3bb}}{2}, \end{align*} $$

where we set $k := b_{\phi (\unicode{x3bb} /2,\varepsilon )} + 1$ .

Now, observe that

$$ \begin{align*} \sum_{n \le \psi(\unicode{x3bb},\varepsilon)} I_{C(a_n,b_n) \cap B_k^c} < \phi(\unicode{x3bb}/2,\varepsilon), \end{align*} $$

and, thus,

$$ \begin{align*} \frac{(\psi(\unicode{x3bb},\varepsilon) + 1)\unicode{x3bb}}{2} \le \sum_{n \le \psi(\unicode{x3bb})} \mu(C(a_n,b_n) \cap B_k^c) = \mu\bigg(\sum_{n \le \psi(\unicode{x3bb})} I_{C(a_n,b_n) \cap B_k^c}\bigg) < \phi(\unicode{x3bb}/2,\varepsilon), \end{align*} $$

which is a contradiction. Here, the last inequality follows from the monotonicity of the integral.