Topological fundamental groupoid. III. Haar systems on the fundamental groupoid (2305.06926v1)

Abstract: Let $X$ be a path connected, locally path connected and semilocally simply connected space; let $\tilde{X}$ be its universal cover. We discuss the existence and description of a Haar system on the fundamental groupoid $\Pi_1(X)$ of $X$. The existence of a Haar system on $\Pi_1(X)$ is justified when $X$ is a second countable, locally compact and Hausdorff. We provide equivalent criteria for the existence of the Haar system on a locally compact (locally Hausdorff) fundamental groupoid in terms of certain measures on $X$ and $\tilde{X}$. $\mathrm{C}^*(\Pi_1(X))$ is described using a result of Muhly, Renault and Williams. Finally, two formulae for the Haar system on $\Pi_1(X)$ in terms of measures on $X$ or $\tilde{X}$ are given.