System of imprimitivity

Remarks

In most uses, $X = G/H$ for a closed subgroup $H$, with $G$ acting by left multiplication.

Example: let $G$ be a discrete group acting on a countable set $X$, let $\mathcal{H} = \ell^2(X)$ with basis $(\delta_x)$, let $\pi(g)\delta_x = \delta_{gx}$, and let $P(E)$ be multiplication by the indicator $\mathbf{1}_E$. For $y \in X$, $\pi(g)P(E)\pi(g)^{-1}\delta_y = \pi(g)P(E)\delta_{g^{-1}y}$, which is $\delta_y$ if $g^{-1}y \in E$ and $0$ otherwise. This is $P(gE)\delta_y$, so clause 3 holds.

Non-example: take $G = X = \Z$ acting by translation, $\mathcal{H} = \ell^2(\Z)$, the same $P$, but $\pi(g) = I$ for all $g$. Then clause 3 would give $P(\set{0}) = P(\set{1})$, which is false.

The induced representation $\operatorname{Ind}_H^G \sigma$ has a canonical system of imprimitivity based on $G/H$, in which $P(E)$ multiplies functions by $\mathbf{1}_E$. For second countable $G$, the Mackey imprimitivity theorem says that every system of imprimitivity based on $G/H$ is equivalent to one of these.