# Characterizations of weak convergence of finite measures and probability measures #

This file will provide portmanteau characterizations of the weak convergence of finite measures and of probability measures, i.e., the standard characterizations of convergence in distribution.

## Main definitions #

The topologies of weak convergence on the types of finite measures and probability measures are already defined in their corresponding files; no substantial new definitions are introduced here.

## Main results #

The main result will be the portmanteau theorem providing various characterizations of the weak convergence of measures (probability measures or finite measures). Given measures μs and μ on a topological space Ω, the conditions that will be proven equivalent (under quite general hypotheses) are:

(T) The measures μs tend to the measure μ weakly. (C) For any closed set F, the limsup of the measures of F under μs is at most the measure of F under μ, i.e., limsupᵢ μsᵢ(F) ≤ μ(F). (O) For any open set G, the liminf of the measures of G under μs is at least the measure of G under μ, i.e., μ(G) ≤ liminfᵢ μsᵢ(G). (B) For any Borel set B whose boundary carries no mass under μ, i.e. μ(∂B) = 0, the measures of B under μs tend to the measure of B under μ, i.e., limᵢ μsᵢ(B) = μ(B).

The separate implications are:

`MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto`

is the implication (T) → (C).`MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge`

is the equivalence (C) ↔ (O).`MeasureTheory.tendsto_measure_of_null_frontier`

is the implication (O) → (B).`MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure`

is the implication (B) → (C).

TODO:

- Prove the remaining implication (O) → (T) to complete the proof of equivalence of the conditions.

## Implementation notes #

Many of the characterizations of weak convergence hold for finite measures and are proven in that generality and then specialized to probability measures. Some implications hold with slightly more general assumptions than in the usual statement of portmanteau theorem. The full portmanteau theorem, however, is most convenient for probability measures on pseudo-emetrizable spaces with their Borel sigma algebras.

Some specific considerations on the assumptions in the different implications:

`MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto`

assumes`PseudoEMetricSpace`

. The only reason is to have bounded continuous pointwise approximations to the indicator function of a closed set. Clearly for example metrizability or pseudo-emetrizability would be sufficient assumptions. The typeclass assumptions should be later adjusted in a way that takes into account use cases, but the proof will presumably remain essentially the same.- Where formulations are currently only provided for probability measures, one can obtain the
finite measure formulations using the characterization of convergence of finite measures by
their total masses and their probability-normalized versions, i.e., by
`MeasureTheory.FiniteMeasure.tendsto_normalize_iff_tendsto`

.

## References #

- [Billingsley,
*Convergence of probability measures*][billingsley1999]

## Tags #

weak convergence of measures, convergence in distribution, convergence in law, finite measure, probability measure

### Portmanteau: limsup condition for closed sets iff liminf condition for open sets #

In this section we prove that for a sequence of Borel probability measures on a topological space and its candidate limit measure, the following two conditions are equivalent:

(C) For any closed set F, the limsup of the measures of F under μs is at most the measure of F under μ, i.e., limsupᵢ μsᵢ(F) ≤ μ(F); (O) For any open set G, the liminf of the measures of G under μs is at least the measure of G under μ, i.e., μ(G) ≤ liminfᵢ μsᵢ(G).

Either of these will later be shown to be equivalent to the weak convergence of the sequence of measures.

One pair of implications of the portmanteau theorem: For a sequence of Borel probability measures, the following two are equivalent:

(C) The limsup of the measures of any closed set is at most the measure of the closed set under a candidate limit measure.

(O) The liminf of the measures of any open set is at least the measure of the open set under a candidate limit measure.

### Portmanteau: limit of measures of Borel sets whose boundary carries no mass in the limit #

In this section we prove that for a sequence of Borel probability measures on a topological space and its candidate limit measure, either of the following equivalent conditions:

(C) For any closed set F, the limsup of the measures of F under μs is at most the measure of F under μ, i.e., limsupᵢ μsᵢ(F) ≤ μ(F); (O) For any open set G, the liminf of the measures of G under μs is at least the measure of G under μ, i.e., μ(G) ≤ liminfᵢ μsᵢ(G).

implies that

(B) For any Borel set B whose boundary carries no mass under μ, i.e. μ(∂B) = 0, the measures of B under μs tend to the measure of B under μ, i.e., limᵢ μsᵢ(B) = μ(B).

One implication of the portmanteau theorem: For a sequence of Borel probability measures, if the liminf of the measures of any open set is at least the measure of the open set under a candidate limit measure, then for any set whose boundary carries no probability mass under the candidate limit measure, then its measures under the sequence converge to its measure under the candidate limit measure.

### Portmanteau implication: weak convergence implies a limsup condition for closed sets #

In this section we prove, under the assumption that the underlying topological space `Ω`

is
pseudo-emetrizable, that

(T) The measures μs tend to the measure μ weakly

implies

(C) For any closed set F, the limsup of the measures of F under μs is at most the measure of F under μ, i.e., limsupᵢ μsᵢ(F) ≤ μ(F).

Combining with a earlier proven implications, we get that (T) implies also both

(O) For any open set G, the liminf of the measures of G under μs is at least the measure of G under μ, i.e., μ(G) ≤ liminfᵢ μsᵢ(G); (B) For any Borel set B whose boundary carries no mass under μ, i.e. μ(∂B) = 0, the measures of B under μs tend to the measure of B under μ, i.e., limᵢ μsᵢ(B) = μ(B).

If bounded continuous functions tend to the indicator of a measurable set and are uniformly bounded, then their integrals against a finite measure tend to the measure of the set. This formulation assumes:

- the functions tend to a limit along a countably generated filter;
- the limit is in the almost everywhere sense;
- boundedness holds almost everywhere.

If a sequence of bounded continuous functions tends to the indicator of a measurable set and the functions are uniformly bounded, then their integrals against a finite measure tend to the measure of the set.

A similar result with more general assumptions is
`MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator`

.

The integrals of thickened indicators of a closed set against a finite measure tend to the measure of the closed set if the thickening radii tend to zero.

One implication of the portmanteau theorem: Weak convergence of finite measures implies that the limsup of the measures of any closed set is at most the measure of the closed set under the limit measure.

One implication of the portmanteau theorem: Weak convergence of probability measures implies that the limsup of the measures of any closed set is at most the measure of the closed set under the limit probability measure.

One implication of the portmanteau theorem: Weak convergence of probability measures implies that the liminf of the measures of any open set is at least the measure of the open set under the limit probability measure.

One implication of the portmanteau theorem: Weak convergence of probability measures implies that if the boundary of a Borel set carries no probability mass under the limit measure, then the limit of the measures of the set equals the measure of the set under the limit probability measure.

A version with coercions to ordinary `ℝ≥0∞`

-valued measures is
`MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto'`

.

### Portmanteau implication: limit condition for Borel sets implies limsup for closed sets #

In this section we prove, under the assumption that the underlying topological space `Ω`

is
pseudo-emetrizable, that

(B) For any Borel set B whose boundary carries no mass under μ, i.e. μ(∂B) = 0, the measures of B under μs tend to the measure of B under μ, i.e., limᵢ μsᵢ(B) = μ(B)

implies

(C) For any closed set F, the limsup of the measures of F under μs is at most the measure of F under μ, i.e., limsupᵢ μsᵢ(F) ≤ μ(F).

Combining with a earlier proven implications, we get that (B) implies also

(O) For any open set G, the liminf of the measures of G under μs is at least the measure of G under μ, i.e., μ(G) ≤ liminfᵢ μsᵢ(G).

One implication of the portmanteau theorem: Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E), then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F).

One implication of the portmanteau theorem: Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E), then for all open sets G we have the limsup condition μ(G) ≤ liminf μsᵢ(G).