Definition 2

Let T be a topological space. One says that T is a Radon (resp. strongly Radon) space if T is Hausdorff and if every function defined on the Borel tribe $\mathscr{B}(T)$ of T, with values in $\overline{\mathbf{R}}_+$, that is countably additive and bounded (resp. locally bounded) is inner regular.

For example, we shall see later on (Prop. 3) that every Polish space is strongly Radon. In particular, every locally compact space with a countable base is strongly Radon.

There exist Radon spaces that are not strongly Radon.