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The following various definitions of a Radon measure seem to be given for the Borel sigma algebra of different types of topological spaces: general, Hausdorff, locally compact, or locally compact Hausdorff.

I was wondering if the definitions are related in some way?

Can these definitions or most of them be unified?

References are appreciated! Thanks and regards!

  1. From Measure Theory, Volumes 1-2 by Vladimir I. Bogachev

    Let $X$ be a topological space. A Borel measure $\mu$ on $X$ is called a Radon measure if for every $B$ in $B(X)$ and $ε>0$, there exists a compact set $K_ε ⊂ B$ such that $|\mu|(B - K_ε) <ε$.

  2. From Wikipedia:

    On the Borel $σ$-algebra of a Hausdorff topological space $X$, a measure is called a Radon measure if it is

    • locally finite, and
    • inner regular.
  3. From ncatlab

    If $X$ is a locally compact Hausdorff topological space, a Radon measure on $X$ is a Borel measure on $X$ that is

    • finite on all compact sets,
    • outer regular on all Borel sets, and
    • inner regular on open sets.
  4. From planetmath

    Let $X$ be a Hausdorff space. A Borel measure $\mu$ on $X$ is said to be a Radon measure if it is:

    • finite on compact sets,
    • inner regular (tight).
  5. From Wikipedia's Radon measures on locally compact spaces

    When the underlying measure space is a locally compact topological space, the definition of a Radon measure can be expressed in terms of continuous linear functionals on the space of continuous functions with compact support.

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