Example

— Let K be a compact subspace of T, i the canonical injection of K into T, and $\lambda$ a measure on K; since i is continuous and $\lambda$ is bounded, i is $\lambda$-proper. Formula (3) of §1, No. 3 shows that the "measure on T defined by $\lambda$" is the image measure $i(\lambda)$.

Remark 5). — If $\theta$ is a real measure and $\pi$ is $\theta$-proper, then $\pi$ is proper for the measures $\theta^+$ and $\theta^-$; one then sets $\pi(\theta) = \pi(\theta^+) - \pi(\theta^-)$. If $\theta$ is a complex measure and $\pi$ is $\theta$-proper, then $\pi$ is proper for the real measures $\Re(\theta)$ and $\Im(\theta)$; one then sets