Let
K
1
,
K
2
{\mathcal {K}_1},\,{\mathcal {K}_2}
be lattices of subsets of a set
X
X
with
K
1
⊂
K
2
{\mathcal {K}_1} \subset {\mathcal {K}_2}
. The main result of this paper states that every semifinite tight set function on
K
1
{\mathcal {K}_1}
can be extended to a semifinite tight set function on
K
2
{\mathcal {K}_2}
. Furthermore, conditions assuring that such an extension is uniquely determined or
σ
\sigma
-smooth at
ϕ
\phi
are given. Since a semifinite tight set function defined on a lattice
K
\mathcal {K}
[and being
σ
\sigma
-smooth at
ϕ
\phi
] can be identified with a semifinite
K
\mathcal {K}
-regular content [measure] on the algebra generated by
K
\mathcal {K}
, the general results are applied to various extension problems in abstract and topological measure theory.