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Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and the strong norm topology, where the norm is the total variation norm.

Surprisingly, I have found very little discussion in the literature comparing these two topologies rigourously, besides the oft-cited claim that the norm topology is much stronger than the weak topology. I am looking for a reference that discusses and compares these topologies, esp. things like convergence, boundedness, open sets, projections, etc.

I am mostly concerned with probability measures $\mathcal{P}(X)\subset\mathcal{M}(X)$, but I am not sure how much of a difference this makes wrt topological concerns.

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    $\begingroup$ I vaguely recall that Richard M. Dudley has something on this somewhere. Try his book Probabilities and metrics. Convergence of laws on metric spaces, with a view to statistical testing. ... or maybe the book Real analysis and probability $\endgroup$
    – Gerald Edgar
    Jun 17, 2019 at 11:58
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    $\begingroup$ Have you looked at Bogachev's Measure Theory? Section 4.6 discusses the norm topology, and Chapter 8 is all about the weak topology. If it doesn't have what you want, could you be more specific about what it is that you want? $\endgroup$
    – Nate Eldredge
    Jun 17, 2019 at 15:16
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    $\begingroup$ @user64494 Can you please stop this kind of cosmetic edit? Let us instead respect the preference of the original poster if the intended meaning is clear. (FWIW, it is "according to", not "up to".) $\endgroup$
    – Yemon Choi
    Jun 17, 2019 at 17:54
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    $\begingroup$ I am not the downvoter, but I can understand the downvote. The Wikipedia page on convergence of measures is quite extensive (e.g convergence is discussed, some claims on openness follow and I don't know what you have in mind with projections). So maybe the reason to vote to close was "unclear what you are asking"... Also, it is pretty straightforward to work out many things directly when you have a specific question on weak vs strong convergence. $\endgroup$
    – Dirk
    Jun 17, 2019 at 18:38
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    $\begingroup$ That said: For me, what tells weak and strong convergence apart is the fact that the total variation distance of two diracs is always two (when they do not agree), but they converge weakly towards each other when the base points do. $\endgroup$
    – Dirk
    Jun 17, 2019 at 18:39

1 Answer 1

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A very good treatment of the subject can be found in "Topological Vector Spaces" by Helmut H. Schaefer.

If you have no previous experience at all with functional analysis, it may be a bit harsh in the beginning though. You can also learn first some more basic functional analysis, for example from the first 3 chapters of the classical book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis (this is the route I have personally followed for example).

I expand on my answer in response to the comment

Note that first that, if $X$ is compact, then all continuous functions on $X$ are bounded. Therefore in this case the "probabilist's weak" topology on the space of finite Radon measures $\mathcal{M}(X)$ is just the weak* topology (in the usual sense of functional analysis) on $C(X)^*$ via the isomorphism $\mathcal{M}(X)\cong C(X)^*$. Therefore the comparison between the two topologies is clear in this case.

This does not work anymore if $X$ is not compact, and here is where the more specific approach of the probabilist comes into play. Let $\mathcal{P}(X)$ be the space of Borel probability on $X$. Then you can define a distance on $\mathcal{P}(X)$ by $$ d_P(\mu,\nu):=\inf\{\alpha>0\,:\,\mu(A)\le \nu(A_{\alpha})+\alpha,\, \mu(A)\le \nu(A_\alpha)+\alpha\},$$ where $$ A_\alpha=\begin{cases} \{x\in X\,:\,d(x,A)<\alpha\} & A\ne \emptyset,\\ \emptyset & A=\emptyset. \end{cases} $$ This is called Prokhorov distance, and you can prove that, provided $X$ being separable, the "probabilist's weak" topology is induced by the metric $d_P$.

If fact, in order to make the comparison you are interested in, it suffices to compare this metric with the one inducing the strong topology.

What makes things more complicated from a functional analitic point of view is that the metric $d_P$ is in general not induced by norm. It follows that the functional analysis you most commonly learn in your studies does not apply. Schaefer's book is all about extending those tools to the case in which a norm to start with is not available.

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  • $\begingroup$ Can you give a more specific reference within the H.H. Schaefer book? This is one of the first books I checked, and I only found one passing reference to total variation on p. 43. (To clarify: I am not referring to the usual norm vs. weak topology in functional analysis, but the corresponding more specific notions in probability theory.) $\endgroup$
    – JohnA
    Jun 17, 2019 at 13:31
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    $\begingroup$ The topology defined by the Prokhorov distance is the weak-* topology on $\mathcal{P}(X)$ considered as a subspace of $C_b(X)^*$ (the dual of the bounded continuous functions). The embedding is $\mu \mapsto \left(f \mapsto \int_X f \mathrm{d}\mu\right)$. $\endgroup$
    – Robert Furber
    Jun 17, 2019 at 20:14

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