On the continuity of the geometric side of the trace formula

We extend the geometric side of Arthur’s non-invariant trace formula for a reductive group $G$ defined over $\mathbb{Q}$ continuously to a natural space $\mathcal{C}(G({\mathbb{A}}^1))$ of test functions which are not necessarily compactly supported. The analogous result for the spectral side was obtained in [MR2811597]. The geometric side is decomposed according to the following equivalence relation on $G(\mathbb{Q})$: ${\gamma }_1\sim {\gamma }_2$ if ${\gamma }_1$ and ${\gamma }_2$ are conjugate in $G(\bar{\mathbb{Q}})$ and their semisimple parts are conjugate in $G(\mathbb{Q})$. All terms in the resulting decomposition are continuous linear forms on the space $\mathcal{C}(G(\mathbb{A}{)}^1)$, and can be approximated (with continuous error terms) by naively truncated integrals.