Show that for $f,g \in L^{1}(G)$ we have $f*g \in L^{1}(G)$.

$\int |g(x^{-1}y)|dy=\int |g(y)|dy$ since you are integrating w.r.t. Haar measure. Hence, By Fubini/Tonelli Theorem we get $\int \int |g(x^{-1}y)|dy|f(x)|dx=\|f\|_1\|g\|_1$.

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Integration

General Topology

Group Theory

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