Example

— 1) If $p$ is $\mu$-proper and $\nu = p(\mu)$, then $\nu$ is a pseudo-image measure of $\mu$ under $p$ (Ch. V, §6, No. 2, Cor. 2 of Prop. 2).

2) Let $B'$ be a locally compact space, $\nu'$ a positive measure on $B'$; take for T the space $B \times B'$ and for $\mu$ the measure $\nu \otimes \nu'$; if $p$ is the projection of T onto B, then $\nu$ is a pseudo-image of $\mu$ under $p$ (Ch. V, §8, No. 2, Prop. 4 and No. 3, Cor. 1 of Prop. 7).

Note that if $\nu$ is a pseudo-image measure of $\mu$ under $p$, then $\nu$ is carried by $p(T)$.

If $\nu$ is a pseudo-image of $\mu$ under $p$, the set of measures that are pseudo-images of $\mu$ under $p$ is the class of positive measures equivalent to $\nu$, and every positive measure equivalent to $\mu$ admits the same pseudo-image measures under $p$. The class of $\nu$ is said to be the pseudo-image class of that of $\mu$ under $p$.