(\[eq:pos\]) holds for every $\phi$ in the dual of $\mathcal{M}(X)$. Now use (\[eq:suma\]), to obtain (\[eq:form\]).

Now we can prove Theorem \[main2\], the main result in this work.

By the hypotheses, whenever $h \in H$ one has $h_{\#} \mu = \mu$. Let $\{ u_n \}$ be a bounded sequence in $M^p_{1,H}(X,\mu)$, namely a bounded sequence in $M^p_1(X,\mu)$ such that $h^{\#} u_n = u_n$ for each $n$ and each $h$ in $H$. Then the sequence of measures $\{ \mu_n \}$ defined by $$\mu_n := \mu \llcorner |u_n|^{p^{\ast}}$$ is also $H$-invariant.

On the other hand, if the sequence $\{ u_n \}$ converges weakly to some $u$ in $M_{1,H}^p(X,\mu)$, then[^9] by Theorem \[theorem:Lions\] there exists a subsequence[^10] of $\{ \mu_n \}$ such that $$\label{measure-inv}
\mu_n \to \mu \llcorner |u|^{p^{\ast}} + \sum_{i \in I} \mu_i \delta_{x_i}$$ in $\mathcal{M}(X)$, where $I$ is at most countable.

In addition, it is not difficult to see that if $\{ \mu_n \}$ is a sequence of $H$-invariant measures converging to some $\nu$ in $\mathcal{M}(X)$, then $\nu$ is also $H$-invariant; therefore from (\[measure-inv\]) the measure $$\mu \llcorner |u|^{p^{\ast}} +  \sum_{i \in I} \mu_i \delta_{x_i}$$ is $H$-invariant. Moreover, since $\mu \llcorner |u|^{p^{\ast}}$ is $H$-invariant, then $\sigma := \sum_{i \in I} \mu_i 