Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a bounded linear functional which is invariant under the action of $Q$, i.e. $m(gf)=m(f)$ for all $f\in\mathcal{A}$, where: $$gf(x):=f(g^{-1}\cdot x)\quad\forall f\in\mathcal{A},g\in Q,x\in X$$ Assume that $m=m_{1}-m_{2}+\mathbf{i} m_{3}-\mathbf{i} m_{4}$, where $m_{1},m_{2},m_{3}$ and $m_{4}$ in $\mathcal{A}^{\ast}$ are positive functionals, i.e. $m_{j}(f)\geq 0$ for all $1\leq j\leq 4$ and $f\geq 0$.
Question: is $m_{j}$ invariant under $G$? It seems somehow obvious but I have not managed to write it down for some reason.
In case you want to know: I am given a locally compact group $G$ and closed subgroups $H,Q\leq G$, a $Q$-invariant functional $m$ on $\mathscr{C}_{b}^{lu}(G/H)$, which are the bounded continuous functions such that the orbit maps $g\mapsto gf$ are continuous. I need $m$ to be normalized and positive, so that for $X:=G/H$ I use $\mathscr{C}_{b}(X)^{\ast}\cong\mathcal{M}(\beta X)$, the space of complex Radon measures on the Stone-Čech compactification. These functionals I can decompose as needed and I want to use the positive part only.