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!
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 $|\mu|(B - K_ε) <ε$.
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.
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.
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).
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.