What is the relation between weak convergence of measures and weak convergence from functional analysis

To keep things simple, we assume X to be a polish space (think of X as Rn for example). Let’s denote with P(X) the space of all Borel probability measure on X. We say {μn}P(X) converges weakly to μP(X), denoted by μnμ if

fdμnfdμ,fCb(X)

where Cb(X) is the space of all bounded continuous real valued functions. Let this definition be seen as the purely probabilistic one.

From functional analysis we also have a concept of weak convergence and weak topology. Let E be a Banach space and denote by E the dual. Then by considering the family {ϕf:fE}, where ϕf:ER is the linear functional ϕf(x):=f,x, the weak topology on E is the coarsest topology which makes all ϕf continuous. One can prove xnx weakly (in weak topology) if and only if ϕf(xn)ϕ(x) for all fE.

Since X is Polish, we have that P(X) is Polish too. Now I can define a continuous linear functional ϕf:P(X)R for fCb(X) by

ϕf(μ):=fdμ

Infact ϕfP(X). Therefore we have Cb(X)P(X).

My questions are

1. Question Is the dual P(X) known? Is it isomorphic to a well known space? Is Cb(X) a proper subspace?

2. Question The notation of weak convergence in the probabilistic sense (μnμ) is weaker than the weak convergence in the functional analytical sense. What I mean is: If μnμ in the weak toplogy, i.e. f,μnf,μ for all fP(X) this implies μnμ since Cb(X)P(X). Is this the reason why one calls μnμ weak convergence, or is there any other reason?

Answer

“Weak convergence of measures” is a misnomer. What it really means is that the space of measures is identified, via Riesz representation, with the dual of some space of continuous functions, and this gives us weak* topology on the space of measures. People just don’t like saying that their measures converge weak-star-ly, or putting a lot of asterisks in their texts.

Folland writes in Real Analysis, page 223:

The weak* topology on M(X)=C0(X) … is of considerable importance in applications; we shall call it the vague topology on M(X). (The term “vague” is common in probability theory and has the advantage of forming an adverb more gracefully than “weak*”.) The vague topology is sometimes called the weak topology, but this terminology conflicts with ours, since C0(X) is rarely reflexive.

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