1. Introduction
Recently, McNicholl and D. Rojas [7] developed a framework to study the effective theory of weak convergence of measures on . Recall that a sequence of finite Borel measures on a separable metric space weakly converges to a measure if, for every bounded continuous realvalued function on ,
. Weak convergence is used extensively in probability theory, particularly in the study of optimization problems in stochastic dynamic programming
[2] and controlled Markov processes [3]. This paper partially serves as a continuation of [7].As seen in [6], the space of finite Borel measures on a computable metric space forms a computable metric space when equipped with the Prokhorov metric. It is well known that the Prokhorov metric, introduced by Prokhorov [9] in 1956, metrizes the topology of weak convergence of measures. Thus, if a uniformly computable sequence in converges effectively in the Prokhorov metric, then the limit is a computable measure. This leads us to the following.
Question 1.1.
Are effective weak convergence and effective convergence in the Prokhorov metric equivalent?
In this paper, we provide a positive answer to this question in the case of . In the classical theory, the equivalence between weak convergence and convergence in the Prokhorov metric uses the Portmanteau Theorem, a characterization theorem for weak convergence originally due to Alexandroff [1]. Thus, the key to answering Question 1.1 is the effective Portmanteau Theorem (Theorem 5.1 in [7]).
Although not as commonly studied, another notion of convergence for sequences of measures in analysis and probability theory is vague convergence. A sequence of (not necessarily finite) Borel measures on a separable metric space vaguely converges to a measure if, for every compactlysupported continuous realvalued function on , . If is a sequence of probability measures, then converges vaguely if and only if it converges weakly (see [4]). This fact is used in [8] to define an effective notion of convergence for probability measures on . However, this definition is not a suitable effective analogue to classical vague convergence for nonprobability measures on . We are thus led to the following.
Question 1.2.
What is a suitable definition of effective vague convergence?
Following [7], we propose two answers to this question, one of which is uniform (Definition 5.2) and one of which is not (Definition 5.1). As in the case of effective weak convergence, we show that these definitions are equivalent (Theorem 5.3). In contrast to effective weak convergence, we also show that effective vague convergence does not guarantee a computable limit even when the limit is finite. Nevertheless, it is possible to obtain a computable limit under effective vague convergence.
As previously stated, classical weak and vague convergence coincide at sequences of probability measures. Along the same vein, we determine the point at which effective weak and vague convergence coincide (Theorem 5.8). This yields an effective version of the correspondence between classical weak and vague convergence for probability measures (Corollary 5.12).
This paper is divided as follows. Section 2 consists of the necessary background in both classical and computable analysis and measure theory. Section 3 covers some preliminary material used in latter sections to state and prove the main results of this paper. In Section 4, we prove the equivalence between effective weak convergence and effective convergence in the Prokhorov metric in . In Section 5, we define effective notions of vague convergence in and analyze the aforementioned consequences. We conclude in Section 6 with a discussion of the results and implications for future research in this direction.
2. Background
2.1. Background from classical analysis
In this paper, we denote the set of all continuous functions on by , the set of all bounded continuous functions on by , and the set of all compactlysupported continuous functions on by . We define the support of a function to be the set .
For and , let . For , we denote the open neighborhood of by . For and , is called the open neighborhood of . We denote the Borel algebra of by . The Prokhorov metric is defined as follows: for any ,
2.2. Background from computable analysis and computable measure theory
We say that a bounded interval is rational if each of its endpoints is rational. Fix an effective enumeration of the set of all rational open intervals.
An open set is if is c.e.. Similarly, a closed set is if is c.e.. We denote the set of subsets of by , and we denote the set of subsets of by . We say that indexes if indexes . Indices of sets in are defined analogously. A pair of sets is indexed by an if is indexed by and is indexed by .
For a compact set , a (minimal cover) name for is an enumeration of all minimal finite covers of . We say is computably compact if it has a computable name. An index of is defined to be an index of a name of .
If is a rational compact interval, let denote the space of polygonal functions on with rational vertices; we will refer to these functions as rational polygonal functions on . When , we may extend to all of by letting for and for . We may also extend to be supported on a computably compact set.
Fix a real number . A (Cauchy) name of is a sequence of rational numbers so that and so that for all .
When is a compact rational interval and is a rational open interval, we let . A (compactopen) name of a function is an enumeration of . If , then is computable if and only has a computable name.
Fix . We say is computable if it has a computable name. An index of such a name is also said to be an index of . We say is leftc.e. (rightc.e.) if its left (right) Dedekind cut is c.e.. It follows that is computable if and only if it is leftc.e. and rightc.e.. A sequence is computable if is computable uniformly in .
A function is computable if there is a Turing functional so that is a name of whenever is a name of . An index of such a functional is also said to be an index of . We denote the set of all bounded computable functions on by . We denote the set of all compactlysupported computable functions on by .
A function is lower semicomputable if there is a Turing functional so that enumerates the left Dedekind cut of whenever is a name of . A function is upper semicomputable if is lower semicomputable.
A function is computable if there is a Turing functional so that is a name of whenever is a name of . An index of such a functional is also said to be an index of .
Each of the names we have just discussed can be represented as a point in for a sufficiently large alphabet .
Suppose is a convergent sequence of reals, and let . A modulus of convergence of is a function so that whenever .
A measure is computable if is a computable real and is leftc.e. uniformly in an index of ; i.e. it is possible to compute an index of the left Dedekind cut of from an index of . A sequence of measures in is uniformly computable if is a computable measure uniformly in .
Suppose is computable. A pair of subsets of is almost decidable if , , and . If, in addition, , then we say that is a almost decidable pair of . We then say is almost decidable if it has a almost decidable pair. Suppose is a almost decidable pair of . Then, indexes if for some index of and some index of . We note that almost decidable sets are effective analogues of continuity sets.
3. Preliminaries
The following effective notions of weak convergence of measures appear in [7].
Definition 3.1.
We say effectively weakly converges to if for every , and it is possible to compute an index of a modulus of convergence of from an index of and a bound on .
Definition 3.2.
We say uniformly effectively weakly converges to if it weakly converges to and there is a uniform procedure that for any computes a modulus of convergence of from a c.o.name of and a bound on .
Observe that Definition 3.2 is uniform, whereas Definition 3.1 is not uniform. Nevertheless, we have the following.
Theorem 3.3 ([7]).
Suppose is uniformly computable. The following are equivalent.

is effectively weakly convergent.

is uniformly effectively weakly convergent.
As we will see in Section 5, we will model the effective definitions of vague convergence in after Definitions 3.1 and 3.2.
In addition, we need the following pair of definitions.
Definition 3.4.
Suppose is a sequence of reals, and let .

We say witnesses that is not smaller than if is the left Dedekind cut of and if whenever and .

We say witnesses that is not larger than if is the right Dedekind cut of and if whenever and .
In order to prove our first main result in the next section, we will need the following theorem.
Theorem 3.5 (Effective Portmanteau Theorem, [7]).
Let be a uniformly computable sequence in . The following are equivalent.

effectively weakly converges to .

From so that indexes a uniformly continuous with , it is possible to compute a modulus of convergence of .

is computable, and from an index of it is possible to compute an index of a witness that is not larger than .

is computable, and from an index of it is possible to compute an index of a witness that is not smaller than .

is computable, and for every almost decidable , and an index of a modulus of convergence of can be computed from a almost decidable index of .
4. Effective Convergence in the Prokhorov Metric
We say that a sequence in converges effectively in the Prokhorov metric to a measure if there is a computable function such that for every , implies . Since forms a computable metric space under , it follows that every uniformly computable sequence of measures in converges to a computable measure in . As metrizes the topology of weak convergence of measures in , it is natural to characterize effective convergence in as an effective notion of weak convergence. However, it is not immediately clear that effective convergence in can be obtained from effective weak convergence and vice versa. The following result states that both of these convergence notions coincide for uniformly computable sequences on measures in .
Theorem 4.1.
Suppose is uniformly computable. The following are equivalent:

is effectively weakly convergent;

converges effectively in .
First, we need the following lemma.
Lemma 4.2.
Let be computable, and let be rational. It is possible to compute an open cover of consisting of open balls with radius less that , each of which is a almost decidable set.
Proof.
Adapt the proof of Lemma 5.1.1 in [6] by replacing with . ∎
The proof of the classical version of Theorem 4.1 makes use of the classical Portmanteau Theorem as well as a classical version of Lemma 4.2. As we shall see below, Theorem 4.1 makes use of the effective Portmanteau Theorem as well as Lemma 4.2. However, before proving Theorem 4.1, we also require the following lemma.
Lemma 4.3.
If , then for any rational .
Proof.
Let be an enumeration of all rational open intervals of . Then, for each , for some with . Let , and let . It suffices to show that is c.e.. Now, for each , we enumerate into whenever . Note that if and only if , which occurs if and only if . Since , is lowersemicomputable (Theorem 5.1.2 in [10]). Thus, the enumeration is effective. Since was arbitrary, the result follows. ∎
Proof of Theorem 4.1.
Suppose that converges effectively in to . Then, is computable, and we have a computable function such that for all and all , . In particular, for any and all ,
By Theorem 3.5, it suffices to compute an index of a witness that is not larger than from an index of .
Fix . Wait until is enumerated into the right Dedekind cut of . Since is computable and by Lemma 4.3, is rightc.e. for any . Search for the first so that . Then, search for the first such that . Let , and let . Therefore, for all ,
It follows that is an index of a witness that is not larger than .
Next, suppose that effectively weakly converges to . Then, is computable. By Theorem 3.5, we can compute for every almost decidable an index of a modulus of convergence of from a almost decidable index of .
We build the function by the following effective procedure. First, let . By Lemma 4.2, we can compute a sequence of uniformly almost decidable rational open balls in with radius less than such that . Search for the first so that . Let
Then, is a finite collection of almost decidable sets. Define to be the smallest index so that for every and every . Thus,
for all .
To see that is the desired function, fix . Let
Then, and satisfies the following properties:

.

.

for all .
Therefore, for all ,
and
Since was arbitrary, it follows that for all , as desired. ∎
Theorem 4.1 serves as further evidence that effective weak convergence is the appropriate computable analogue to weak convergence. When viewing as a computable metric space, effective weak convergence can be defined as the effective topology induced by in .
5. Effective Vague Convergence in
In Sections 3 and 4, we discussed effective weak convergence in . In this section, we effectivize the definition of vague convergence. Just as in the case of weak convergence, we provide a nonuniform definition (Definition 5.1) and a uniform definition (Definition 5.2).
Definition 5.1.
We say effectively vaguely converges to if for every , and it is possible to compute an index of a modulus of convergence of from an index of and an index of .
Definition 5.2.
We say uniformly effectively vaguely converges to if it weakly converges to and there is a uniform procedure that for any computes a modulus of convergence of from a name of and a name of .
As expected, the following theorem established the equivalence between these two definitions.
Theorem 5.3.
Suppose is uniformly computable. The following are equivalent.

is effectively vaguely convergent.

is uniformly effectively vaguely convergent.
Before we prove this theorem, we need the following lemma.
Lemma 5.4.
For all , it is possible to compute a rational polygonal function so that from a name of and a name of .
Proof.
Since and are computable, we may compute rationals and . Now, compute a function that approximates on the interval with the property that . ∎
Proof of Theorem 5.3.
It is possible to compute a name of and a name of from an index of and an index of . It thus follows that every uniformly effectively vaguely convergent sequence is effectively vaguely convergent.
Now, suppose effectively vaguely converges to . Let be a name of , and let be a name of . We construct a function as follows. Let , which can be computed from by Lemma 5.2.6 in [10]. Let be a function on given by
Note that , and so . Since effectively vaguely converges to , we can compute an index so that whenever . It follows that for every , .
By Lemma 5.4, we can compute a rational polygonal function with the property that and
Since effectively vaguely converges to , we can compute an so that whenever . Set .
Suppose . Then,
Thus, is a modulus of convergence of . Since the construction of from and is uniform, uniformly effectively vaguely converges to . ∎
By the same reasoning as with effective weak convergence, we may also conclude the following.
Corollary 5.5.
Suppose is a uniformly computable sequence in that is effectively vaguely convergent. Then, is vaguely convergent.
Recall that effective weak limit measures are computable. It is reasonable to ask if this translates to effective vague convergence as well. Below, we provide a negative answer to this question.
Proposition 5.6.
There is a uniformly computable sequence of measures that effectively vaguely converges to a limit measure with the property that is incomputable.
Proof.
Let be an incomputable c.e. set, and let be an effective enumeration of . For each , let . Note that is a uniformly computable sequence of measures. We will show that effectively vaguely converges to the measure .
For starters, fix and an index of . Then, we can compute by Lemma 5.2.6 in [10]. Observe that for all ,
Thus, is an index of a modulus of convergence of the sequence . Finally, note that is incomputable since it is the limit of a Specker sequence. ∎
We have in the proof of Proposition 5.6 above not only an example of an incomputable effective vague limit, but also a finite one. This leads us to ask the following question: when are effective vague limit measures computable? We provide a necessary and sufficient condition for which this is the case.
Proposition 5.7.
Suppose is a uniformly computable sequence in that effectively vaguely converges to . If is computable, then is computable.
Proof.
Suppose is computable. It suffices to show that is leftc.e. for every uniformly in an index of .
For starters, let be a rational open interval. Since is nonnegative and lowersemicomputable, we can compute a sequence of computable Lipschitz functions such that increases to pointwise and for each (see Proposition C.7, [5]). By the Monotone Convergence Theorem, for any .
Fix . Since effectively vaguely converges to , we can compute an index of a modulus of convergence of with for each from an index of and an index of . This means is a computable real for each . Thus, we enumerate into the left Dedekind cut of if we can find such that and . It follows that is leftc.e. uniformly in an index of .
Now, fix . Then, can be expressed as a countable union of rational open intervals. By the observation above, it follows that is the limit of an increasing sequence of leftc.e. reals. Therefore, is leftc.e. uniformly in an index of . ∎
Another way of ensuring that effective vague limits are computable is by analyzing the point in which effective weak and vague convergence coincide. Below, we provide a sufficient condition for which these notions do coincide.
Theorem 5.8.
Suppose is uniformly computable. Suppose further that there is a computable modulus of convergence of . The following are equivalent.

is effectively vaguely convergent.

is effectively weakly convergent.
In fact, we will prove that effective vague convergence is equivalent to uniform effective weak convergence. The following series of lemmas will allow us to carry out a proof of Theorem 5.8 similar to the proof of Theorem 3.3.
Lemma 5.9.
Suppose is uniformly computable and effectively vaguely converges to . Suppose further that there is a computable modulus of convergence of . For every such that , and it is possible to compute an index of a modulus of convergence of from an index of and an index of .
Proof.
Fix and an index of . By Theorem 5.3, we can compute a modulus of convergence for . By assumption, there is a computable modulus of convergence for . Therefore, is an index of a modulus of convergence of . ∎
Lemma 5.10.
Suppose is uniformly computable and effectively vaguely converges to . Suppose further that there is a computable modulus of convergence of . From , it is possible to compute such that for all .
Proof.
Combine Lemma 2.5 and the proof of Lemma 4.5 in [7]. ∎
Lemma 5.11.
Suppose is uniformly computable and effectively vaguely converges to . Suppose further that there is a computable modulus of convergence of . From a name of an and an so that , it is possible to compute and so that is computably compactly supported, , and whenever .
Proof.
We modify the proof of Lemma 4.6 in [7] so that is computably compactly supported. Then, use a name of and a name of to compute with the desired properties. ∎
Proof of Theorem 5.8.
It is immediate that implies . Now, suppose effectively vaguely converges to . Fix with name and bound . We construct the function as follows. By means of Lemma 5.11, we can compute and so that is computably compactly supported, , and whenever . Since effectively vaguely converges to , we can compute an so that whenever . Set .
Suppose . Then,
Thus, is a modulus of convergence of . Since the construction of from and is uniform, uniformly effectively weakly converges to . The result follows by Theorem 3.3. ∎
We conclude this section by deriving from Theorem 5.8 the following effective version of a classical result in probability theory.
Corollary 5.12.
Suppose is a uniformly computable sequence of probability measures. The following are equivalent.

is effectively vaguely convergent.

is effectively weakly convergent.
6. Conclusion
We expanded the effective framework for the study of weak convergence of measures in introduced in [7] by demonstrating the equivalence between effective weak convergence and effective convergence in the Prokhorov metric. This provides further evidence that effective weak convergence is the appropriate analogue to classical weak convergence in . While the Prokhorov metric is useful in defining as a computable metric space, effective weak convergence is a more useful tool to analyze properties of as a computable metric space. Theorem 4.1, therefore, unifies the approaches in [6] and [7] to studying the effective theory of weak convergence in .
Additionally, we introduced two effective notions of vague convergence in . While the moduli of convergence in the first definition are produced for computable functions in , moduli of convergence in the second definition are produced for all functions in via names. Similar to effective weak convergence, Theorem 5.3 shows that they are equivalent. Just as in the classical sense, however, there are notable differences between effective weak convergence and effective vague convergence.
Consider the following classical example. The sequence of point masses converges vaguely to the zero measure, but it does not converge weakly since for any supported on . This distinction carries over in the effective setting, albeit in a more computationally significant manner. In Proposition 5.6, we gave an example of a uniformly computable sequence of measures that effectively vaguely converges to a finite incomputable measure. Thus, the “vagueness” of effective vague convergence is present in the fact that limits under this convergence notion may not be computable even when finite.
Nevertheless, we provide evidence that effective vague convergence is the appropriate computable analogue to classical vague convergence. For instance, we found in Theorem 5.8 a sufficient condition for which effective weak and vague convergence coincide. Consequently, Corollary 5.12 provides the following observation: whereas classical weak and vague convergence coincide at the probability measures, effective weak and vague convergence coincide at the computable probability measures. Since we argue that effective weak convergence is the appropriate computable analogue to classical weak convergence, a similar argument follows in the case of effective vague convergence. In the future, we would like to generalize the definitions of effective weak and vague convergence to measures in for an arbitrary computable metric space .
7. Acknowledgements
We would like to thank Timothy McNicholl for proofreading and providing several valuable comments and suggestions.
8. Declarations
The author did not receive support from any organization for the submitted work. The author has no financial or proprietary interests in any material discussed in this article. Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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