Remark

— 1) When T is locally compact, $\mu$ is the image of $\lambda$ under the injection of L into T. We shall see in §2, No. 3, Example, when image measures will have been treated, that this interpretation remains valid for arbitrary spaces. 2) We shall also see that the measures defined in Example 2 are positive measures on T with compact support (No. 6, Remark 2)).

We shall henceforth consider only positive measures, absent express mention to the contrary. For the rest of this section, T will denote a topological space and $\mu$ a positive measure on T.

Numerous results in the following subsections may be extended to positive premeasures. This extension is left to the reader.