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I'm reading Philip Protter's "Stochastic integration and differential equations", just had a quick question about it. He defines L^0 as "the space of finite-valued random variables topologized by convergence in probability". I'm having a little trouble working out what this means. I took a brief course in Metric Spaces in college, but never got any further than that. The wikipedia page says that a topology defines the open sets of a space. How does this apply here? Is it that the open sets in L^0 are given by all finite-valued rv's and all limits in probability of finite-valued rv'?If anyone could give a brief explanation of how space can be topologized by some mode of convergence, or link to a book, that would be really helpful.

Hi,If I remember correctly, in general the set of convergent sequences does not uniquely determine a topology (this can be proven). In the case of convergence of probability though, it is "well-known" that the functional d(X,Y)=E(f(X,Y)) where f(x,y)=|y-x| / (1 + |y-x|), defines a metric on the space of (classes of) real-valued random variables, and the associated convergence is the right one.

Nice one moltabile.Any chance of a reference where I could learn more about this?