TITLE: A net version of dominated convergence?
QUESTION [19 upvotes]: 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 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.

REPLY [3 votes]: Here is a nice theorem that might interest you ("‎Methods of Modern Mathematical Physics‎: ‎Functional analysis‎", ‎Volume 1‎, ‎Michael Reed‎, ‎Barry Simon)‎:

‎Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$‎. ‎Let $(f_\alpha)$ be an increasing net of continuous functions. Then $f=\lim_\alpha f_\alpha \in L^1(X,\mu )$ if and only if $\sup_\alpha \| f_\alpha \|_1 < \infty$, and in this case $\lim_\alpha \| f-f_\alpha\|_1 = 0$‎.