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Weak-star topology - what is it?

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    I'm doing an Ergodic Theory module, and the weak-star topology seems to be coming up a lot but I just can't seem to get my head around it.

    Here's how it's defined in the course: http://www.warwick.ac.uk/~masdbl/erg...ture-first.pdf (page 6)

    Let X be a compact metric space... Let \mathcal{M} be the space of probability measures on X. We can consider the weak-star topology on \mathcal{M} corresponding to a sequence (\mu_n)\in\mathcal{M} converging to \mu\in\mathcal{M} (i.e. \mu_n\to\mu precisely when \int f d\mu_n \to \int f d\mu for every f\in C(X)
    Now here, I'm not entirely sure how they're using these sequences of measures to define a topology (I mean, what do the open sets look like?). So, I go and look it up on the internet and I get the definition below which is about functionals this time, rather than measures.

    Definition from http://planetmath.org/encyclopedia/W...rTopology.html

    Let X be a locally convex topological vector space (over \mathbb{C} or \mathbb{R}) and let X^* be the set of continuous linear functionals on X (the continuous dual of X). If f\in X^* ... let p_x(f) denote the seminorm p_x(f)=|f(x)|. The topology on X^* defined by the seminorms \{p_x|x\in X\} is called the weak-* topology.
    So.. if we're considering these seminorms, would an open ball (centre f, radius delta) look something like \{ g \in X^* | \forall x \in X : p_x(g-f) < \delta \} = \left\{ g \in X^* \middle| \forall x \in X : |g(x)-f(x)| < \delta \right\}? I'm a bit confused here because I've only ever seen open balls for one particular norm on a set - not many different seminorms. Then presumably the topology would be generated by these open balls.

    So is anyone able to please explain what these definitions mean and how they're connected?
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    (Original post by ttoby)
    I'm doing an Ergodic Theory module, and the weak-star topology seems to be coming up a lot but I just can't seem to get my head around it.

    Here's how it's defined in the course: http://www.warwick.ac.uk/~masdbl/erg...ture-first.pdf (page 6)



    Now here, I'm not entirely sure how they're using these sequences of measures to define a topology (I mean, what do the open sets look like?). So, I go and look it up on the internet and I get the definition below which is about functionals this time, rather than measures.

    Definition from http://planetmath.org/encyclopedia/W...rTopology.html



    So.. if we're considering these seminorms, would an open ball (centre f, radius delta) look something like \{ g \in X^* | \forall x \in X : p_x(g-f) < \delta \} = \left\{ g \in X^* \middle| \forall x \in X : |g(x)-f(x)| < \delta \right\}? I'm a bit confused here because I've only ever seen open balls for one particular norm on a set - not many different seminorms. Then presumably the topology would be generated by these open balls.

    So is anyone able to please explain what these definitions mean and how they're connected?
    Okay I've read some stuff on wikipedia and now I'm going to hit you up with some (probably very) naive thoughts:

    On wiki the weak * topology on X is defined to be the coarsest one making all the functionals in X* continuous.

    Wiki

    In our case X is our measure space. SO if we assume (here's hoping) all our functionals on our measure space are

    \nu \mapsto \int f d\nu for f \in C[X]

    then the topology defined in the course is going to be the one that does this. I'm not sure how this works in with the planetmath definition but this is one way of thinking about the course definition in terms of functionals.
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    planetmath seems to be down, which sucks. I'm a bit confused because the planetmath definition is trying to define a topology on X*, when we're looking for a topology on X.
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    (Original post by Hathlan)
    Okay I've read some stuff on wikipedia and now I'm going to hit you up with some (probably very) naive thoughts:

    On wiki the weak * topology on X is defined to be the coarsest one making all the functionals in X* continuous.

    Wiki

    In our case X is our measure space. SO if we assume (here's hoping) all our functionals on our measure space are

    \nu \mapsto \int f d\nu for f \in C[X]

    then the topology defined in the course is going to be the one that does this. I'm not sure how this works in with the planetmath definition but this is one way of thinking about the course definition in terms of functionals.
    Thanks, yeah it helps to think of it like that. I won't worry about the planet math definition then.

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