Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. Define the function from $G$ to $\mathcal{L}^1(G)$ that assigns to each $y$ in $G$ the function that takes $x$ in $G$ to $f(xy^{-1})$. I.e. it assigns the $y$-translate of $f$. Is this a continuous function of $y$ into $\mathcal{L}^1$? 

On p.94 3.10 of [these notes](http://www.math.uwaterloo.ca/~lwmarcou/Preprints/PMath810Notes.pdf) it is asserted that it is. The usual proof of this sort of thing in cases where $G=\mathbb{R}$, say, is you use density of compactly supported continuous functions, and then dominated convergence. The former survives, but dominated convergence does not work on nets. In fact, a limit of a net of measurable functions need not be measurable.