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I've heard about possibility to introduce measure to some topological spaces, without knowing the nature of its elements. Is it true? I doubt, if this is true for arbitrary topological space (unless I'm satisfied with some trivial measures like counting one or constant-zero-measure), however if it is possible in some cases, then are there any books or papers about it?

I've read in here that it was an old question, and there are some useful remarks here... though I'm more interested in reading a bit more about it, especially if I'd like to quote some scientific lectures in my work.

If anyone could provide me with some resources on that topic - I'd be grateful.

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I_Really_Want_To_Heal_Myself
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    This may be of interest, for the special case where the topology is given by a metric. induced measure – Andres Mejia Apr 03 '17 at 11:24
  • Well, unfortunately I'm working mostly on non-metrizable spaces, but yes, this might be helpful as well. Anything else would be nice. – I_Really_Want_To_Heal_Myself Apr 03 '17 at 11:49
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    Well, here is what I would think of as the basic consideration:

    Whenever we are given a topology, there is a collection of open sets. These give rise to a borel $\sigma$-algebra generated by open sets. This has the benefit that:

    1. Closed and open sets are in your $\sigma$-algebra
    2. The continuous functions are all measurable.

    I'm certainly no expert, but after a quick google search, it appears that this construction is quite common on probability theory.

     – Andres Mejia Apr 03 '17 at 12:00
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    The construction of measures on locally compact Hausdorff spaces is a pretty standard topic in real analysis and measure theory, I think. For instance, I know that Royden's and Folland's real analysis textbooks cover this. This book by Fremlin on this topic has been recommended to me as well. Also, you could just look at wikipedia on Radon measure. – aduh Apr 03 '17 at 14:16
  • See "Haar Measure" (note the spelling) in Wikipedia. It is a measure on a locally compact, Hausdorff topological group. The space $\mathbb R^n $ ($1\leq n<\infty$) where the group operation is vector addition, is such a group – DanielWainfleet Apr 03 '17 at 19:05

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Volume 4 of Measure Theory, by Fremlin is a great resource. The TeX sources can be downloaded and compiled, or you could buy them as well. All books are "encyclopaedic" (quite thorough) and cover a lot of standard and more obscure results. One part of volume 4 is about the construction of a nice measure on certain types of spaces (like on locally compact Hausdorff spaces, using the dual of $C(X)$ and other functional analysis techniques...) and also on the interplay between a measure and topology (often minimally all open sets should be measurable, so we have Borel measures), for some spaces such measures sometimes we get so-called regularity in different forms (inner / outer etc.). You should check them out..

aduh
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Henno Brandsma
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