= Solution
The printed symbol $E$ is not defined. Take $E=L^1(\mathbb R)$, or more generally an $L^1$ subspace with its induced $L^1$ <norm>; density must be in that <norm>. Given $g\in E$, choose a sequence $f_n\in F$ with
$$
\|g-f_n\|_1\leq2^{-2n}.
$$
By the assumed convergence on $F$, each $T_rf_n\to T_0f_n$ holds outside a <null set> $N_n$. Their <countable union> $N=\bigcup_nN_n$ is still null. Thus outside $N$ the convergence statements for every member of this single approximation sequence hold simultaneously. No assertion of simultaneous convergence for the entire uncountable dense class is needed.
Back to article page
