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A probability space consists of a sample space, X, which is a set of all possible random elements. A collection of random events which is a sigma field, F, which consists of all the events that belong to X. A probability measure that assigns a number between [0,1] for each event in the sigma field F and statisfies the axioms of probability.

A topological space is a set, X, such that a topology, T, has been specified (under the 3 main topological properties).

I read that: Topological spaces are used to define a notion of "closeness". With it, you can intuitively speak about points which are close to each other. (However, we may not know how close: this is a metric space). …. A measure space serves an entirely different goal. A measure space is made to define integrals… Reference https://www.physicsforums.com/threads/measurable-spaces-vs-topological-spaces.558644/

A metric space is a metrizable space X with a specific metric d that gives the topology of X. Therefore, there is a connection between a metric space and a topological space namely a topological space may be induced by a metric space. Is that true? In this case the connection between topological spaces and measureable spaces can be changed to be: What is the relationship between measurable spaces and metric spaces a question that has some relevant answer in What's the relationship between a measure space and a metric space?

But I still do not see a well-written answer to that question and how is it connected to topological spaces with a defined metrics? Any resources or help is appreciated Thank you very much in advance

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If we have a metric space $(X,d)$ we can define a topology (the smallest one that makes all metric balls open). This topology generates a Borel $\sigma$-algebra (the smallest one that contains the topology). On the latter we can talk about measures. It's a solved problem in general topology to characterise those topological spaces that can be defined from a metric (the so-called metrisable topological spaces, look up the Bing-Nagata-Smirnov theorem, among others).

But the metric can also give rise to a metric outer measure directly and the resulting measure (via the standard Carathéodory construction) is at least defined on all Borel sets (as above). I think the Hausdorff measure for $s=1$ gives the Lebesgue measure on the reals, I think. I'm not sure whether the analogous problem of recognising such spaces among all measure spaces is open or not.

If we have a measure space $(X,\Sigma, \mu)$ with finite measure, we can define a metric on the set of equivalence classes of $\Sigma$ under the equivalence relation $\sim$ defined by $A \sim B$ iff $\mu(A \Delta B) = 0$, where the latter is the symmetric difference $A \Delta B = (A \setminus B) \cup (B \setminus A)$, by $d([A],[B]) = \mu(A \Delta B)$. A measure is called separable whenever this metric space (or rather, its topology) is separable.

This about exhausts what I know about metric/measure relations. Maybe it helps.

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