Beschreibung:
<jats:p>Let <jats:italic>S(G)</jats:italic> be a Segal algebra on an infinite compact Abelian group <jats:italic>G</jats:italic>. We study the existence of many discontinuous translation invariant linear functionals on <jats:italic>S(G)</jats:italic>. It is shown that if <jats:italic>G</jats:italic>/<jats:italic>C<jats:sub>G</jats:sub></jats:italic> contains no finitely generated dense subgroups, then the dimension of the linear space of all translation invariant linear functionals on <jats:italic>S(G)</jats:italic> is greater than or equal to 2<jats:italic><jats:sup>C</jats:sup></jats:italic> and there exist 2<jats:italic><jats:sup>C</jats:sup></jats:italic> discontinuous translation invariant linear functionals on <jats:italic>S</jats:italic>(<jats:italic>G</jats:italic>), where <jats:italic>c</jats:italic> and <jats:italic>C<jats:sub>G</jats:sub></jats:italic> denote the cardinal number of the continuum and the connected component of the identity in <jats:italic>G</jats:italic>, respectively.</jats:p>