Stabilizer of probability measure on projective space

Question

I'm reading notes on Lattices on Lie groups (in French: Réseaux des groupes de Lie) by Yves Benoist and I have some troubles to understand some aspects on the setting & proof of the Lemma 7.3 (due to Furstenberg) on page 60.

Let $k= \mathbb{R}, \mathbb{C}$ or a finite extension of $p$-adic field $\mathbb{Q}_p$, $p$ prime. Let $E:=k^d$ be a vector space and $\mathbb{P}(E)$ the associated projective space, $\operatorname{PGL}(E)$ the group of projective transformations on $\mathbb{P}(E)$ and $\nu$ a probability measure on $\mathbb{P}(E)$.

Let $S:= \{ g \in \operatorname{PGL}(E) \ \vert \ g_* \nu = \nu \} $.

(recall $ g_* \nu $ is the push-forward or image measure with respect automorphism $g: \mathbb{P}(E) \to \mathbb{P}(E)$)

Questions:

1) In the notes there is not a word said about the $\sigma$-algebra with which the projective space $\mathbb{P}(E)$ is endowed which allows to consider a (probability) $\nu$. In most cases when dealing with measures on affine space $k^d$ without making any comment on the underlying $\sigma$-algebra, one uses in silence the Borel-$\sigma$-algebra, a kind of canonical $\sigma$-algebra on affine spaces.

Is there any type of "canonical" $\sigma$-algebra for projective spaces which is in most cases used when one is discussing measures on projective spaces without mentioning by name the underlying $\sigma$-algebra? So is there a kind of standard $\sigma$-algebra for projective spaces known as a sort of pendant to the Borel-$\sigma$-algebra for affine spaces?

2) At the beginning of the proof of the Lemma there is remarked that it's rather obvious that the stabilizer $S:= \{ g \in \operatorname{PGL}(E) \ \vert \ g_* \nu = \nu \} $ of probability measure $\nu$ is a closed subspace in $\operatorname{PGL}(E)$.

Why that's the case? I assume that the topology on $\operatorname{PGL}(E)$ is the quotient topology induced from $\operatorname{GL}(E) \subset \operatorname{End}(E) \cong k^{d^2}$.

It is known that if in general a topological group $G$ acts continuously on a topological space $S$ by continuous map $a: G \times S \to S$, then any stabilizer is closed. But here we consider the "stabilization" of a measure und there is no standard way I know to endow the "space of meausures" on $\mathbb{P}(E)$ with continuous $\operatorname{PGL}(E)$-action.

How the closedness of $S= \{ g \in \operatorname{PGL}(E) \ \vert \ g_* \nu = \nu \} $ can be deduced? I not see the exact reason why that's really clair at all.

Answer 1

1) The canonical projection $\pi:E\setminus0\to \mathbb{P}(E)$ is a (topological) quotient map; the $\sigma$-algebra of $\mathbb{P}(E)$ is the associated Borel $\sigma$-algebra (in your situation, and in general if $\sigma$-algebras are not mentioned). By an argument similar to that discussed at https://math.stackexchange.com/a/4348985/169085 this is the same thing as saying $B\subseteq \mathbb{P}(E)$ is measurable iff $\pi^{-1}(B)\subseteq E\setminus0$ is Borel measurable.

2) Note that $\mathbb{P}(E)$ is a compact metric space. Consequently the space $\mathcal{M}(\mathbb{P}(E))$ of Borel probability measures on $\mathbb{P}(E)$ is compact w/r/t the weakstar topology. Further, the action $\mathbb{P}\text{GL}(E)\curvearrowright \mathbb{P}(E)$ is by homeomorphisms, so the action $\mathbb{P}\text{GL}(E)\curvearrowright \mathcal{M}(\mathbb{P}(E)), g\curvearrowright \mu=g_\ast(\mu)$ is by affine homeomorphisms. (You are correct to assume $\mathbb{P}\text{GL}(E)$ to be endowed with the quotient topology). Now you can use the general statement about stabilizers of topological actions, or alternatively you can take $g_n\to g$ and argue that if $(g_n)_\ast(\nu)=\nu$, then $g_\ast(\nu)=\nu$ also. Of course, the stabilizer is not only a closed subspace; it's a closed subgroup.